harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{\underline X}=\partial_{\underline x}+\partial_{\underline y}=\sum_{j=1}^pe_j\partial_{x_j}+\sum_{j=1}^qe_{p+j}\partial_{x_{p+j}}, \Delta_{\underline X}=\Delta_{\underline x}+\Delta_{\underline y}=\sum_{j=1}^p\partial_{x_j}^2+\sum_{j=1}^q\partial_{x_{p+j}}^2.
\int_{I_{m,1}}B_{p,q}'(x)\dx =\int_{\frac{1}{p^{m+1}(p-1)}}^{\frac{1}{p^{m+1}}}B_{p,q}'(x)\dx &=B_{p,q}\left(\frac{1}{p^{m+1}}\right)-B_{p,q}\left(\frac{1}{p^{m+1}(p-1)}\right) &=\frac{1}{q^{m+1}}-\frac{1}{q^{m+1}(q-1)}&=\frac{q-2}{q^{m+1}(q-1)}
\dfrac{Q^{(n-2)}(0)}{(n-2)!}=3\left( n-3\right) n\left( n^{2}-9n+2\right) -5(n-2)\left( n+1\right) \left( n^{2}-13n+6\right) -\dfrac{1}{2}(n+2)(n+1)^{2}n(n-1) =-\dfrac{1}{2}\left( n-5\right) \left( n+2\right) p(n)
I_2 = \int_{\Omega_{\ell}}\Big[W_1'(x_2) - \delta x_1\{\rho_{\ell}W_1''(x_2) + W_1'(x_2)\rho'_\ell(x_2)\}\Big]^2 &= \int_{\Omega_{\ell}}|W_1'|^2 -2\delta\int_{\Omega_{\ell}}x_1W_1'(x_2)\big\{\rho_{\ell}W_1''(x_2) + W_1'(x_2)\rho'_\ell(x_2)\big\}&\phantom{=}+\delta^2 \int_{\Omega_{\ell}}x_1^2| \rho_{\ell}W_1''(x_2) + W_1'(x_2)\rho_\ell'(x_2)|^2.
V_k^{j,\tau}(T,\bar{s}) =O(e^{(\max\{\lambda_{i},\lambda_1 a\}-\lambda_1+8\tau(1+\lambda_1))r(0,k)}) W_k^{j,\tau}(T,\bar{s}) =O(e^{(\max\{\lambda_{i}-\lambda_1,\lambda_1 a-\lambda_1,-\lambda_j\}+9\tau(1+\lambda_1))r(0,k)}).
\frac{(z-1)^{n+k}}{(1+z)^k}e^{ktz}= \frac{e^{kt}}{2^k}(z-1)^{n+k}\sum_{m \geq 0}C_m(-k,-2kt)\frac{(kt(z-1))^m}{m!}& = e^{kt}\sum_{m \geq 0}L_{m}^{(-k-m)}(-2kt)\frac{(z-1)^{m+n+k}}{2^{m+k}}.
\begin{array}{c c c}{\partial^{\mu}C_{n \mu}^{F}= 0 ,}&{\quad \partial_{\mu}b_{n}^{F \mu}= 0 ,}&{\quad \partial_{\mu}\bar{C}_{n}^{F \mu}= 0 ,}\\{\partial^{\mu}C_{n \mu}^{B}= 0 ,}&{\quad \partial_{\mu}b_{n}^{B \mu}= 0 ,}&{\quad \partial_{\mu}\bar{C}_{n}^{B \mu}= 0 .}\\ \end{array}
\varphi(y) ( e^{-t L_1^{(p)}}1_x,1_y ) = (e^{-t L_1^{(p)}}1_x,U^*U 1_y) &= ( U e^{-t L_1^{(p)}}1_x,U 1_y ) &= ( e^{-t L_2^{(p)}}U1_x,U 1_y).
\mathrm{P}^o_{m,2}\approx 1- \sum^{N}_{n=1}\bar{w}_ne^{-\frac{c_{n,\mathcal{R}_s}\frac{1}{\rho}}{\tilde{\tau}}}q\left(\frac{c_{n,\mathcal{R}_s}}{\tilde{\tau}}\right) ,
T_{a}^{L}= \left( \begin{array}{c c c}{N_{a}}&{E_{a}^{\dagger}}&{F_{a}^{\dagger}}\\{E_{a}}&{P_{a}}&{G_{a}^{\dagger}}\\{F_{a}}&{G_{a}}&{b_{a}}\\ \end{array}\right) , v = \left( \begin{array}{c}{u}\\{w}\\{z}\\ \end{array}\right) ,
\left(\sum_{i = 1}^{m}\kappa_i + m\right)\int_{\Omega}\prod_{i = 1}^{m}\vert y_i\vert^{\kappa_i + 1}dz\leq C_{21}\| (y_1, y_2, \cdots, y_m) \|_{Y,1}^{\frac{\sum_{i = 1}^{m}\kappa_i + m}{p}}.
U_{_W}(t_i,t_{i-1}) = \frac{1}{2\pi}\int d\triangle_i d\xi_i\exp\{-i(\xi_i x_i+\triangle_ip_i)\}\textrm{Tr}[\hat{U}(t_i,t_{i-1})\exp i(\xi_i \hat{x}+\triangle_i\hat{p})].
\kappa_{\theta}:= \Big(\frac{(\theta+1)(\theta-2)}{2}+ 2 \theta L \Big) \ \textrm{and}\K_{\theta}:= 2^{\theta-1}L{}^{\theta}\Big( 1 + \theta + \frac{p(p-1)}{2}\Delta^{\frac{\theta}{2}-1}\Big) \mathbb E \vert Z \vert^{\theta}, ~ Z \sim{\cal N}(0;I_d).
S_{\rm{QH}}=\int_{\cal M}d^3x~\left(\psi^\dagger\,i\left(\partial_0-iA_0-iea_0\right)\psi+\frac{1}{2m_e}\psi^\dagger\left(\nabla-iA-iea\right)^2\psi\right)+\frac{1}{m}S_{\rm{CS}}^{[U(1)]}[A]
A_{1}(\textbf{u}) = \left( \begin{array}{ccc}v_{1}0 0 f(\textbf{u}) v_{1}0 0 0 v_{1}\end{array}\right), \A_{2}(\textbf{u}) = \left( \begin{array}{ccc}v_{2}0 0 0 v_{2}0 f(\textbf{u}) 0 v_{2}\end{array}\right),
\big|\left<\rho_{S,\mathcal{O}}(\beta^* * \beta)\phi_{x,r},\psi_i\right>\big|=\big|\left<\phi_{x,r}, \rho_{S,\mathcal{O}}(\beta^* *\beta)\psi_i\right>\big|=\lambda_i \big|\left<\phi_{x,r}, F_\pi\right>\big|.
h_{n}=\frac{n!\,(\alpha+\beta+\gamma+\delta-1)_{n}(\alpha+\beta)_{n}(\alpha+\gamma)_{n}(\alpha+\delta)_{n}(\beta+\gamma)_{n}(\beta+\delta)_{n}(\gamma+\delta)_{n}}{(\alpha+\beta+\gamma+\delta)_{2n}}.
\begin{array}{ll}\int_D \lambda \, \tau \, dx + \int_D \nabla u \cdot \nabla \tau \, dx = - \int_D \psi_g \, \tau \, dx \forall \ \tau \in H^1_0(D) [6pt]\int_D \nabla \lambda \cdot \nabla v \, dx [4pt]= - \int_D f \, v \, dx - \int_D \nabla \psi_g \cdot \nabla v \, dx + \langle h,v\rangle_{-\frac12}\forall \v \in H^1(D) \cap{\cal H}^\bot\, .\end{array}
a(\phi) = \frac{\phi -4}{(k-1) \phi - k^2}, b(\phi) = \frac{(c-1)(k-1)\phi+(k-c)^2}{(d-1)(c-1)(k-1)\phi+d(k-c)^2+2(c-1)(k-c)}&=\frac{X_\phi}{(d-1)X_\phi+Y},
\|e^{t(\Delta+V)}\|_{L^2(\mathbb{R}^d), L^2(\mathbb{R}^d)}\leq \|t^{-\frac{d}{2}}e^{c_3(t+1)}e^{-\frac{|x|^2}{c_4t}}\|_{L^1(\mathbb{R}^d)}&=(c_4\pi)^{d/2}e^{c_3(t+1)}.
F_a =xy^a+\sum_{s=1}^{a-1}(a-1-s+p+q)xy^{a-s}F_s+pqxF_a &+2\sum_{s=2}^ax\frac{y^{a+1-s}-y^a}{1-y}F_s+\sum_{s\geq a+1}x(2\frac{y-y^a}{1-y}+p+q)F_s+\sum_{s\geq a+2}(s-1-a)xF_s.
y(t)=\int_{-\infty}^{t}h(t-t^{\prime})x(t^{\prime})dt^{\prime}=\int_{-\infty}^{\infty}h(t-t^{\prime})x(t^{\prime})dt^{\prime}=\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau=\int_{0}^{\infty}h(\tau)x(t-\tau)d\tau
\frac{2N(\sigma,T)}{T}\le 2\left[C_1(\sigma)T^{(5-8\sigma)/3}\log^{5 - 2\sigma}T + \frac{C_2(\sigma)\log^2 T}{T}\right], &\le 2\bigg[C_1(\sigma)\exp\left(\frac{B_2(5-8\sigma)}{3}\sqrt{\log x}\right)\left(B_2\sqrt{\log x}\right)^{5 - 2\sigma}&\qquad\qquad\qquad+ C_2(\sigma)\exp\left(-B_2\sqrt{\log x}\right)(B_2)^2\log x\bigg] &= s_2'(x, \sigma),\ .
\begin{array}{l}{{S_{+}\rightarrow d^{2}S_{+}+ 2 c d S_{0}- c^{2}S_{-}\, ,}}\\{{S_{0}\rightarrow b d S_{+}+ ( 1 + 2 b c ) S_{0}- a c S_{-}\, ,}}\\{{S_{-}\rightarrow - b^{2}S_{+}- 2 a b S_{0}+ a^{2}S_{-}\, .}}\\ \end{array}
h_\epsilon(y) = f_{\tiny \mbox{BEP}}^{\delta=0}(y) (1+ \gamma_1 - \gamma_2)+ f_{\tiny \mbox{BEP}}^{\delta=2}(y)(\gamma_2- 2 \gamma_1)+ f_{\tiny \mbox{BEP}}^{\delta=4}(y)\gamma_1+ O(\epsilon^3),
\tilde{\pi}^{ij}=2\gamma^{1/2}\tilde{\gamma}^{-1/3}e^{-\frac{3}{2}(\phi-\tilde{\phi})}\left[ \frac{\partial H}{\partial \tilde{h}_{ij}}-\frac{1}{3}\frac{\partial H}{\partial \tilde{h}_{kl}}\tilde{h}_{kl}\tilde{h}_{ij}\right] ~~~,
A_i-A_i^*= \sum_{0<|h|<H}\left( 1-\frac{|h|}{H}\right) \hat{f}(h) \left( e(h Y_i \alpha )-e(h (Y_i \alpha - \delta_i )) \right) \sum_{k \in H_i}e(h (S_k-Y_i) \alpha ) ,
\langle s e^{\i \langle\theta,\cdot\rangle_{\mathbb{C}^d}}\alpha, e^{\i \langle\theta,\cdot\rangle_{\mathbb{C}^d}}\beta \rangle =\sum_{i,j\in\{1,\ldots,n\}}\left( \int_{Y}s_{ij}\right)\alpha_i \beta_j, ( \alpha, \beta \in \mathbb{C}^n).
M= \begin{bmatrix}3 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 2 2 1 0 0 0 0 1 1 1 1 0 0 0 0 4 0 0 0 3 1 0 0 1 1 0 0 1 1 0 0 0 3 1 3 2 3 2 1 1 1 2 1 0 4 1 1 \end{bmatrix}
S^{\prime}= v \theta \left[ \frac{1}{4}\left\{y \left. \frac{\delta^{2}V ( y )}{\delta y^{2}}\right|_{y = y_{c}}\right\}^{2}- i \chi \psi \left. \frac{\delta^{2}V ( y )}{\delta y^{2}}\right|_{y = y_{c}}\right] .
h \left( D_1 \cdots D_n \right) = \sum_{i=1}^n h(D_i) = \sum_{i=1}^n h_i \iota \left( D_1 \cdots D_n \right) = \sum_{i=1}^n \iota(D_i) + \sum_{1 \leq j < k \leq n}m \left( h_k, \partial h_j \right),
I^\psi_{B(i)}= \sum_{u \in \psi_{B(i)}}g_{u,B(i)}\ell(r_{u,B(i)}) P_B, I^\varphi_{B(i)}= \sum_{v \in \varphi_{B(i)}}g_{v,B(i)}\ell(r_{v,B(i)}) P_M,
\langle\pi|\Delta^{jj'}_{\mu\mu'\nu\nu'}|\pi'\rangle=\, G_{\mu\nu\mu'\nu'}\prod_{\bar s=0}^{2\pi}\,\delta^4(\pi(\bar s)-\pi'(\bar s))\int_{s_-}^{s_+}\!ds'\,ds''\,{{\beta_\epsilon(s'-s)\beta_\epsilon(s''-s)}\over{\pi'_\alpha(s')\pi^{'\alpha}(s'')}},
\frac{1}{(q)_N}\sum_{n = 1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^{n - 1}nq^{n(n + 1)/2}}{1 - q^n}+ \sum_{n = 1}^{N}\left[\begin{matrix}N n \end{matrix}\right] \frac{q^{n^2}}{(q)_n}\sum_{k=1}^n \frac{q^k}{(1-q^k)^2}= \frac{1}{(q)_N}\sum_{n=1}^{\infty}\frac{n q^n(1- q^{N n})}{1- q^n}.
\sup_{\tau< t < S_{\lambda'}+\tau}\int_{{\bf R}^N}e^{-\frac{2\lambda'H_0(y)^2}{1-4\lambda'( t -\tau)}}u_{m,n}(y, t )^2\,dy \le C\int_{{\bf R}^N}e^{- 2\lambda' H_0(y)^2}u_{m,n}(y,\tau)^2\,dy \le C\tau^{-\frac{N}{2}}I_{m,n}^2
F_{mn}^a =- 4\rho^2 \frac{\overline\eta_{mn}^a}{(x^2+\rho^2)^2}+ 8\rho^2\frac{(\overline\eta_{mk}^ax_n-\overline\eta_{nk}^ax_m)x_k}{x^2(x^2+\rho^2)^2}~.
\left ( \begin{array}{c}\varphi_{0}(x) \vdots \varphi_{p}(x) \end{array}\right ) = \sum_{k \in \mathbb{Z}}A_{k}\left ( \begin{array}{c}\varphi_{0}(2x - k) \vdots \varphi_{p}(2x - k) \end{array}\right )
P(T, x) = \prod_{j=1}^{l(T)}\mathcal{G}_j^{n(T, j)}= \prod_{j=1}^{l(T)}\left( \dfrac{W_{l(T)+1-j}}{W_{l(T)-j}}\right)^{n(T, j)}= \dfrac{\displaystyle\prod_{j=1}^{l(T)}W_{l(T)+1-j}^{n(T, j)}}{\displaystyle\prod_{j=1}^{l(T)}W_{l(T)-j}^{n(T, j)}}.
g_{1}(\nabla^{^{M_1}}_{U}V,X) =g_{1}(V,\nabla^{^{M_1}}_{U}\phi \mathcal{B}X)+g_{1}(V,\nabla^{^{M_1}}_{U}\omega \mathcal{B}X)+g_{1}(\mathcal{C}X,\mathcal{H}\nabla^{^{M_1}}_{U}\varphi V)+g_{1}(V,\varphi U)\eta (X).
\mathbb{P}\left( N(Q_{j_1,j_2, \dots,j_{k-1}, j_k})\leq \max(\frac{N(Q)}{(1+c)^{l}},N_0) \right)\geq \sum\limits_{i=l}^{k}C^{i}_k\left(\frac{1}{2A}\right)^{k-i}\left(1-\frac{1}{2A}\right)^i
\begin{pmatrix}\varphi begin{align*}0.3em]\psi \end{pmatrix}=\left(\frac{1}{(2\pi)^2}\int_{BZ}\mathcal{A}^\alpha(\omega,\delta)^{-1}d\alpha \right)\begin{pmatrix}-f begin{align*}0.3em]-g \end{pmatrix}
A_k \left( \overrightarrow{\eta}( \overrightarrow{X},t),t \right) = \frac{1}{2}\hat{A}_l ( \overrightarrow{X},t) \left(\delta_{kl}+ \frac{e_{kl}( \overrightarrow{X},t)}{1 + e\theta \widehat{B}}\right)
\bar{\Delta}\Delta = f^{- 1}1_{[ 2 ] \times [ 2 ]}= \left( \! \begin{array}{c c}{f_{1}^{- 1}}{0_{[ 2 ] \times [ 2 ]}}{0_{[ 2 ] \times [ 2 ]}}{f_{2}^{- 1}}\end{array}\! \right) ,
\begin{array}{c}\theta^{(l)}_0 = \inf\bigl\{t \ge0, X^{(l)}_t \in C\bigr\}, begin{align*}3pt]\theta^{(l)}_n = \inf\bigl\{t \ge\theta^{(l)}_{n-1}, X^{(l)}_t \in C \bigr\},\end{array}
\sum_{\substack{m = -p_1 + p_2 - p_3 p_j \in I_j p_1 \equiv a_1 \pmod{k^2}p_2 \equiv a_2 \pmod{k^2}}}1 = \frac{\mathcal{G}(m)\mathfrak{S}(m)}{(\log X)^3}\cdot \mathbf{1}_{(k, -a_1 + a_2 - m) = (k,a_1) = (k,a_2) = 1}\cdot \frac{f((k, m)) g(k)}{k \varphi(k)^3}+ o \Big ( \frac{X^2}{(\log X)^3}\Big ),
\int_DU'(Y(T,x,z))Y(T,x,z)dx= \tilde{p}(0,z) \exp \big( - \int_0^T \{\Phi(s,z) +\frac{b_0(s,z)}{\sigma_0(s,z)}\}dB(s) &+\frac{1}{2}\int_0^T\{\Phi^2(s,z) -\frac{b_0^2(s,z)}{\sigma_0^2(s,z)}\}ds \big)=:\tilde{p}(0,z)\Gamma(T,z) .
\lim_{q\rightarrow\infty}\|f\|_{L^{\Psi_q}(\mathbb{R},dx)}\geq\liminf_{q\rightarrow\infty}\|f\|_{L^{\Psi_q}(\mathbb{R},dx)}=\liminf_{q\rightarrow\infty}\Psi_q^{-1}(1)^{-1}=\infty>1= \|f\|_{L^\infty(\mathbb{R},dx)}.
a_i(\hat{x}+\hat{y})^{p\lambda_i-1}= \sum_{i'=0}^{r-1}\sum_{j'=0}^{s-1}\alpha_{i,i',j'}\left(\sum_{i''=0}^{p-1}(-1)^{i''}\hat{x}^{i''}\hat{y}^{p-1-i''}\right)\hat{x}^{pi'}\hat{y}^{pj'}.
a^{n+1}_{SUPG}(u_h,v_h) = \epsilon(\nabla u_h, \nabla v_h)_{\Omega_{t^{n+1}}}+ (\mathbf{b}\cdot \nabla u_h, v_h)_{\Omega_{t^{n+1}}}+ (c u_h, v_h)_{\Omega_{t^{n+1}}}+\sum_{K \in \mathcal{T}_{t^{n+1}}}\delta_{K}(- \epsilon\Delta u_h + (\mathbf{b-w}^{n+1}_h) \cdot\nabla u_h + cu_h, ~(\mathbf{b-w}^{n+1}_h ) \cdot \nabla v_h)_K.
\| \nabla \phi^x\|_{L^2(\Omega)}^2 = \langle \phi^x, \varphi^x \rangle_{L^2(\Omega)}\leq \| \phi^x \|_{L^2(\Omega)}\| \varphi^x \|_{L^2(\Omega)}\leq \frac{\| \nabla \phi^x \|_{L^2(\Omega)}}{\sqrt{\lambda_1(\Omega)}}\| \varphi^x \|_{L^2(\Omega)},
\|v(x_1, \cdot)\|_{L^2((0,1))}\leq N \Big( \|v\|_{L^2((x_1 -1, x_1) \times (0,1))}+ \|\nabla v\|_{L^2((x_1 -1, x_1) \times (0,1))}\Big) & \leq N \| \nabla v\|_{L^2((x_1 -1, x_1) \times (0,1))},
\partial_{t}S(t -\frac{t_{n}^{j}}{\lambda_{n}^{j}})(\frac{1}{\lambda_{n}^{j}}u_{0}^{n}(\frac{x}{\lambda_{n}^{j}}), \frac{1}{(\lambda_{n}^{j})^{2}}u_{1}^{n}(\frac{x}{\lambda_{n}^{j}}))|_{t = 0}\rightharpoonup \phi_{0}^{j}(x)
\frac{\sigma^{n+1}- 2 \sigma^n + \sigma^{n-1}}{\tau^2}= \frac{\delta^{n+\frac{1}{2}}- \delta^{n-\frac{1}{2}}}{\tau}\ , \delta^{n+ \frac{1}{2}}+ \delta^{n- \frac{1}{2}}= 2\, \frac{\sigma^{n+ \frac{1}{2}}- \sigma^{n- \frac{1}{2}}}{\tau}.
\left(\partial_t-{l\over 2\pi \alpha'}\partial_\theta\right)^2X_4- \left({l\,\rho \over 2\pi \alpha'}\right)^2 \left({1\over \rho^2}\partial_\theta^2 +\partial_z^2\right) X_4 +O(l^3) =0 \,,
\zeta(2m+1)+2\sum_{n=1}^{\infty}\frac{1}{n^{2m+1}(e^{2\pi n}-1)}=\pi^{2m+1}2^{2m}\sum_{j=0}^{m+1}\frac{(-1)^{j+1}B_{2j}B_{2m+2-2j}}{(2j)!(2m+2-2j)!}.
M_{\Xi}(z,\lambda,w,\mu)=\begin{bmatrix}1 \frac{\overline{w}\,\overline{c(\mu)}}{1-\overline{d(\mu)}\,\overline{w}}\end{bmatrix}\frac{I-\Xi(\mu)^*\Xi(\lambda)}{1-\overline{\mu}\lambda}\begin{bmatrix}1 \frac{zc(\lambda)}{1-d(\lambda)z}\end{bmatrix},
\mathcal{G}_{D,m}(z,s) := -\Gamma\biggl(\frac{s}{2}+1\biggr)\sum_{n|m}\biggl(\frac{D}{n}\biggr)\bigg|\frac{m}{n}\bigg|\sqrt{D}P_{\frac{3}{2}, -\frac{m^2D}{n^2}}\biggl(z,\frac{s}{2}+ \frac{1}{4}\biggr).
\sum_{\rho}\hat{g}(\rho)-\int_{0}^\infty \Big\{g(x)-\frac{1}{4}g^*(x)\Big\}dx +\sum_{n=1}^\infty \Lambda(n)g(n) &+\int_1^\infty \Big\{g(x)+g^*(x)-\frac{2}{x}g(1)\Big\}\frac{x\dx}{2(x^2-1)}+\frac12(\log 4\pi +\gamma)g(1).
\Psi [\phi] = \frac{{1}}{[{\rm \det}(2\pi \hbar G)]^{1/4}}{\rm \exp}\left [ - \int_{x,y}( \phi_{a}(x)-{\hat{\phi}}_a (x) ) \frac{{G_{ab}^{-1}(x,y)}}{4 \hbar}(\phi_{b}(y)-{\hat{\phi}}_b (y) ) \right ] ,
\int_0^RH^{n-1}(A\cap \{dist(\cdot, \partial C)=r\})dr &\le 2^{1/2}(\int_0^RrH^{n-1}(A\cap \{dist(\cdot, \partial C)=r\})dr)^{1/2}&\cdot (\sup_{0<r<R}H^{n-1}(A\cap \{dist(\cdot,\partial C)=r\}))^{1/2},
|\partial_l g \circ f|^2 \leq \left[\sum_{i=1}^3 \frac{\partial g}{\partial f_i}\frac{\partial f_i}{\partial x_l}\right]^2 \leq [3C_2(g)^{\frac{1}{2}}\lambda_2(f)^{\frac{1}{2}}]^2 \leq 9 C_2(g)\lambda_2(f).
Var[\overline{x}(t)]=E[(\overline{x}(t)-E[\overline{x}(t)])^{2}]=\frac{1}{N^{2}}E[(\sum_{k=1}^{N}n(t,k))^{2}]=\frac{1}{N^{2}}\sum_{k=1}^{N}E[n(t,k)^{2}]=\frac{\sigma^{2}}{N}
\sum_{i=1}^{n+1}\frac{1}{s_i}= \sum_{i=1}^n \frac{1}{s_i}+ \frac{1}{s_{n+1}}= 1 - \frac{1}{\prod_{i=1}^n s_i}+ \frac{1}{s_{n+1}}& = 1 - \frac{s_{n+1}- \prod_{i=1}^n s_i}{\prod_{i=1}^{n+1}s_i}= 1 - \frac{1}{\prod_{i=1}^{n+1}s_i}.
a_{N_R,0}=\mu_R,a_{0,1}=\frac{N_R(1-\mu_R)-N_T\mu_T}{N_T(N_R-1)},a_{0,N_R}=\frac{N_T\mu_T+\mu_R-1}{\binom{N_T}{N_R}(N_R-1)}, \textrm{and others being 0,}
\sum_{k=0}^{\infty}(-1)^k\frac{(\frac{1}{2})_k^4(d)_k^3(1-d)_k^3}{(1)_{2k}^3(\frac{1}{2}+d)_{2k}(\frac{3}{2}-d)_{2k}}\Omega_k(d)=\frac{1-2d}{\pi}\tan(d\pi),
\sum_{g\in G}x_{js,g}u_g =\phi(jsu_{k_0})=\phi(ju_{k_0}\alpha_{k_0}^{-1}(s)) &=\sum_{g\in G}x_{j,g}u_g\phi(\alpha_{k_0}^{-1}(s))=\sum_{g\in G}x_{j,g}\alpha_g(\phi(\alpha_{k_0}^{-1}(s)))u_g,
W_{<m,\ell}^\infty(\xi-\xi',x-y)=\frac{4\pi}{\ell^3}\sum_{\substack{|k|_\infty< m k\in \mathbb{Z}^3}}\frac{1}{\left|\frac{2\pi k}{\ell}-(\xi-\xi')\right|^2}\,e^{i\left(\frac{2\pi k}{\ell}-(\xi-\xi')\right)\cdot (x-y)}.
\mathcal{R}(\alpha)=\mathbb{E}\left[{{\rm BW}}(\alpha) \log_2 \left(1+{\rm SINR}(\alpha) \right)\right] \stackrel{(i)}{=}\int\limits_{0}^{\infty}\mathbb{P}\left\{{{\rm BW}}(\alpha) \log_2 \left(1+{\rm SINR}(\alpha) \right)>t \right\}dt, &\stackrel{(ii)}{=}\int\limits_{0}^{\infty}\left[1- \mathcal{O}(\alpha, 2^{\frac{t}{{\rm BW}(\alpha)}}-1) \right] dt,
C_{q,x}^\epsilon(t)=\int_Y\psi_q^\epsilon(y-t)d\rho_x(y)=\int_{y>t+\epsilon}(y-t-\epsilon)^qd\rho_x(y)+\int_{y<t-\epsilon}(t-y-\epsilon)^qd\rho_x(y),\x\in X.
\begin{aligned}2\beta|p|^{2-\alpha}+a(1-\beta)^{\alpha-1}(\beta-\alpha\beta) (\beta-\alpha\beta-1)r^{\beta(1-\alpha)+1}&\leq 2L^{2-\alpha}+a(1-\beta)^{\alpha-1}(\alpha-\alpha^2)(\alpha-\alpha^2-1)\min\{c_0^{\alpha(1-\alpha)+1}, c_0^{2-\alpha}\}\end{aligned}
\varphi^{1}=q^{2}\delta(\varphi^{1}) =q^{2}+T\omega^{2}\delta^{2}(\varphi^{1}) =q^{2}+2T\omega^{2}+T^{2}\left(a_{1}\sin(q^{2})+a_{2}\cos(q^{2})+b_{2}\cos(q^{3})u^{1}\right)
P\left(Q_{\{a,p\}}(x)\right)=-2^{-4/3}\frac{\left(x\sqrt{1-x^2}\right)^{2/3}}{\sigma(x)}=-\frac{x^{2/3}(\sqrt{1-x^2})^{2/3}}{2^{4/3}}Q'_{\{a,p\}}(x),
K_{ij}:=\sum_{k=1}^{H_0-1}(a_{ik}b_{kj}-a^0_{ik}b^0_{kj})+a_{iH_0}b_{H_0j}-a^0_{iH_0}b^0_{H_0j}+\sum_{k=H_0+1}^{H-1}a_{ik}b_{kj}+ a_{iH}b_{Hj}, L_j:= \sum_{k=1}^{H_0-1}(a_{Mk}b_{kj}-a^0_{Mk}b^0_{kj})+a_{MH_0}b_{H_0j}-a^0_{MH_0}b^0_{H_0j}+\sum_{k=H_0+1}^{H-1}a_{Mk}b_{kj}+ a_{MH}b_{Hj}.
{\cal D}^{- 1}= \left( \begin{array}{c c}{{2 \beta p^{2}\theta_{\mu \nu}+ 2 \left( 1 + \frac{m \xi^{2}}{2 \pi}\right) p^{2}\omega_{\mu \nu}}}&{{- \frac{i m \xi \alpha}{2 \pi}p_{\mu}}}\\{{\frac{i m \xi \alpha}{2 \pi}p_{\nu}}}&{{p^{2}- \mu^{2}- \frac{m \alpha^{2}}{\pi}}}\\ \end{array}\right) .
\frac{\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}}{2n\binom{2n}n}=-\frac{16^{n-1}}2\sum_{k=0}^{n-1}\binom{n-1}{k}^2\binom{-\frac12+k}{2n-1}&=\frac18\sum_{k=0}^{n-1}\binom{n-1}k^2\frac{(-1)^{k}(2k)!(4n-2k-2)!}{k!(2n-1)!(2n-k-1)!}.
\sigma^{(n)}_{\mbox{\footnotesize\boldmath $\alpha $}}(z)=\vartheta\left[\begin{array}{c}\alpha_2 /n +1/2 \alpha_1 /n+1/2 \end{array}\right](z,\tau ), ~~~~~~\theta^{(j)}_n (z)=\vartheta\left[\begin{array}{c}1/2 - j/n 1/2 \end{array}\right](z,n\tau ),
{\frac{1}{g_{1}^{2}( q^{2})}}-{\frac{1}{g_{2}^{2}( q^{2})}}\simeq \mathrm{b d . ~ t e r m s}+{\frac{1}{2 4 \pi^{2}}}\ln \left({\frac{\sqrt{q^{2}}}{k}}\right) .
F^{\prime\mu\nu}=\partial^{\prime\mu}A^{\nu}-\partial^{\prime\nu}A^{\mu}=[\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\ -E_{x}/c&0&B_{z}&-B_{y}\\ -E_{y}/c&-B_{z}&0&B_{x}\\ -E_{z}/c&B_{y}&-B_{x}&0\end{matrix}]
\frac{\prod_{i=1}^{n}(1-t^i)}{\prod_{i=1}^{k}(1-t^i)\prod_{i=1}^{n-k}(1-t^i)}=\frac{\prod_{i=n-k+1}^{n}(1-t^i)}{\prod_{i=1}^{k}(1-t^i)}
\beta_{1, j}(R/I_{\Delta}) = \begin{cases}q^2 + q + 1 j = 2 begin{align*}.5ex]\binom{q^2 + q + 1}{3}- \binom{q+1}{3}(q^2 + q + 1) j = 3 begin{align*}.5ex]0 ,\end{cases}
\nabla_X\alpha_{\mathfrak{g}}=\nabla'_X\alpha_{\mathfrak{g}}+[\xi_X,\alpha_{\mathfrak{g}}], &\nabla_X\alpha_{\mathfrak{m}}=[\xi_X,\alpha_{\mathfrak{m}}]_{\mathfrak{g}}+\big( \nabla'_X\alpha_{\mathfrak{m}}+[\xi_X,\alpha_{\mathfrak{m}}]_{\mathfrak{m}}\big).
\hat{f}(s) = \mathcal{M}[f (x); s] = \int_{0}^{\infty}x^{s-1}f (x) dx, f (x) = \mathcal{M}^{-1}[\hat{f}(s); x] = \frac{1}{2\pi i}\int_{L}x^{-s}\hat{f}(s) ds,
\#\left(\{x\in \mathbb{Z}_q\mid s_{h}x=0\}\bigcup\{x_1,\ldots ,x_{h-1}\}\right) = \#\{x\in \mathbb{Z}_q\mid s_{h}x=0\}+\#\{x_1,\ldots ,x_{h-1}\}-\#\left(\{x\in \mathbb{Z}_q\mid s_{h}x=0\}\bigcap\{x_1,\ldots ,x_{h-1}\}\right).
\rho_m(x,t) = \frac1{(4 \pi |t|)^{m/2}}\exp\left( -\frac{|x|^2}{4|t|}\right), \Phi_m(x) = \rho(x,-1) = (2\pi)^{-m/2}\exp\left( -\frac14|x|^2 \right).
\check{R}^{( 2 3 )}( z_{1}- z_{2}) \check{R}^{( 1 2 )}( z_{1}) \check{R}^{( 2 3 )}( z_{2}) = \check{R}^{( 1 2 )}( z_{2}) \check{R}^{( 2 3 )}( z_{1}) \check{R}^{( 1 2 )}( z_{1}- z_{2}) .
\frac{dy_i}{dt}\rho_j+\frac{dy_j}{dt}\rho_i = \frac{a_ix_iy_i}{1+x_i}\rho_j-d_iy_i\rho_j-d_jy_j\rho_i=0 \Rightarrow y_j^* = \frac{\rho_jy_i^* (-a_ix_i^*+d_ix_i^*+d_i)}{d_j\rho_i (x_i^*+i)}
Id_0=\begin{array}{c}Id_0\leftrightarrow\varepsilon \updownarrow \varepsilon\leftrightarrow Id_0\end{array}=\begin{array}{c}Id_0 \updownarrow \varepsilon\end{array}\leftrightarrow\begin{array}{c}\varepsilon \updownarrow Id_0\end{array}=\varepsilon\leftrightarrow\varepsilon=\varepsilon
A_{0}(\textbf{u})\partial_{t}\textbf{u}+ \sum_{j=1}^{d}\textbf{P}A_{0}A_{j}(\textbf{u})\partial_{x_{j}}\textbf{u}+[A_{0}(\textbf{u}), \textbf{P}]A_{j}(\textbf{u})\partial_{x_{j}}\textbf{u}=0.
S = \left( \begin{array}{c c}{{h}}&{{s}}\\{{- s^{\dagger}}}&{{- h^{T}}}\\ \end{array}\right) , \quad V = \left( \begin{array}{c c}{{h^{\prime}}}&{{a}}\\{{- a^{\dagger}}}&{{\left( h^{\prime}\right)^{T}}}\\ \end{array}\right) .
~~\left(\frac{k-1}{2k-1}+\left(\frac{k-1}{2k-1}\right)^2 + \cdots \right) \binom{n}{(k+r+1)/2}+ \binom{n}{r}&< \frac{k-1}{k}\binom{n}{(k+r+1)/2}+ \frac{k-1}{2k-1}\binom{n}{(k+r+1)/2}< \binom{n}{k}.
\begin{aligned}\sum_{t=1}^{T}\sum_{i=1}^{n}\tilde{x}_{it}\tilde{v}_{it}=\sum_{t=1}^{T}\sum_{i=1}^{n}\tilde{x}_{it}v_{it}-\frac{1}{T}\sum_{t=1}^{T}\sum_{i=1}^{n}\sum_{s=1}^{T}\tilde{x}_{it}{v}_{is}. \end{aligned}
u(t) =\mbox{available capacity [Ah] at time $t$}, u(0)=N begin{align*}1mm]\widetilde u(t) =u(t)/N= \mbox{state of charge at time $t$},\widetilde u(0)=1 begin{align*}1mm]E(t) =\mbox{voltage [V] at time $t$},E(0)=E_0.
\int{d^d q\over (2\pi)^d}{1\over q^2 + m_1^2}{1\over (q-k)^2 + m_2^2}={1\over (4\pi)^d}\big\{{2\over \epsilon}- \gamma - b(k^2,m_1^2,m_2^2) +{\cal O}(\epsilon)\big\}
\begin{array}{ll}\Gamma^{\nu}_v(t,x,v;s,y)=G_{\nu}(t,x,v;s,y) +\sum_{k=1}^{\infty}\int_0^t\int_{{\mathbb R}^{2d}}G_{\nu}(t,x,v;\sigma,z)L^{G_{\nu}}_{FO_k}(\sigma,z,s,y)dzds,~z,y\in{\mathbb R}^{2d},\end{array}
F(\lambda) = x^{2\ell-\mu+\lambda}_+\int^\infty_0 (\rho+1)^{2\ell-\mu}\rho^{\lambda-1}d\rho = \frac{\Gamma(\lambda)\Gamma(-2\ell+\mu-\lambda)}{\Gamma(\mu-2\ell)}\, x^{2\ell-\mu+\lambda}_+\,,
k_1(c_i^a v^i_b + c_i^d B^i_j v^j_d \delta_b^a) + k_2 \delta_b^a (c_i^d v^i_d + k c_i^d B^i_j v^j_d \delta_b^a) = c_i^a v^i_b + c_i^d \tilde{B}^i_j v^j_d \delta_b^a.
\Delta=\{x_1x_5x_6,x_2x_4x_6,x_1x_4x_6,x_2x_3x_6,x_1x_2x_6,x_3x_4x_5,x_1x_4x_5,x_2x_3x_5,x_1x_3x_5,x_2x_3x_4\}
\begin{array}{ccll}\dot{x}_1 = -x_1^2-x_2^2-x_3^2 - 2\eta x_1x_2 + x_1 \dot{x}_2 = - 2x_1x_2 - \eta x_1^2 - \eta x_2^2 +\eta x_3^2 - \beta x_3 + x_2 \dot{x}_3 = - 2x_1x_3 -2\eta x_2 x_3 +\beta x_2 +x_3\end{array}
A' \ = \ 2^{n-1}\sum_{\omega = 0}^{a-1}2^{-\omega}(-1)^{\omega}I(\omega, 0) \sum_{\alpha = 0}^\omega \sum_{\delta = \omega - \alpha}^{a - 1 - \alpha}\sum_{i= 0}^{a - \delta - \alpha - 1}(-1)^i \binom{n}{\delta +\alpha}\binom{\delta + \alpha}{\alpha}\binom{n - \delta - \alpha}{i}\binom{\delta}{\omega - \alpha}.
\textstyle[X^jY^{p-j-1-s}]\Psi x^iy^{r-i}\textstyle= \sum_{u,v \in \mathbb F_p}(-1)^{j+s}{p-1-s \choose j}u^{i+p-j-1-s}v^{r-i+j}\textstyle= (-1)^{j+s}{p-1-s \choose j}\sum_{u \in \mathbb F_p}u^{i+p-j-1-s}\sum_{v \in \mathbb F_p}v^{r-i+j}\textstyle= (-1)^{j+s}{p-1-s \choose j}\,\Xi_{i,j},
\frac{1}{g^{2}}\left(\begin{array}{l}\Gamma_{1}\Gamma_{2}\Gamma_{3}\end{array}\right) =\left(\begin{array}{lll}B_{11}B_{12}B_{13}B_{21}B_{22}B_{23}B_{31}B_{32}B_{33}\end{array}\right) \left(\begin{array}{l}\Gamma_{1}\Gamma_{2}\Gamma_{3}\end{array}\right) .
d_{i}^-(\lambda)=\frac{(\lambda_i+1)i(\lambda_i+n-i+[\lambda_i]_2(p-n))}{(-1)^{\sum_{\alpha=i}^n(\alpha+1)(\lambda_\alpha-\lambda_{\alpha+1})}}\bigg(\prod_{\ell=i+1}^n\frac{\lambda_i-\lambda_\ell-i+\ell-1}{\lambda_i-\lambda_\ell-i+\ell-[\lambda_i-\lambda_\ell]_2}\bigg).
\begin{aligned}B(t)=\int_{0}^{t}e^{-(t-s)}g_{}(s)ds=\int_{0}^{t}e^{-(t-s)}(g_0+g_1N(s))ds. C(t)=2\int_{0}^{t}e^{-2(t-s)}a(s)ds=2\int_{0}^{t}e^{-2(t-s)}(a_0+a_1N(s))ds.\end{aligned}
\beta_n(x,\delta) = \int_{|x-t| \ge \delta}\cal K_n(x,t) dt \le \int_{|x-t| \ge \delta}\left( \frac{x-t}{\delta}\right)^2 \cal K_n(x,t) dt & \le \frac{1}{\delta^2}\int_0^1 (x-t)^2 \cal K_n(x,t) dt = \frac{1}{\delta^2}M_n ((\cdot - x)^2)(x) = \frac{1}{\delta^2}\frac{2[ (n-3)x(1-x) + 1]}{(n+2)(n+3)}.
[\varphi(a\otimes x),b\otimes f] = \varphi(a)b \cdot f(x) = \varphi( a\varphi^{-1}(b) \cdot f(x)) = ap^{-1}\varphi^{-1}(b)f(x) &= [a\otimes x, p^{-1}\varphi^{-1}(b \otimes f)].
\int_{0}^{T}e^{\left( T-s\right) A}G \psi_1'(s) = \int_{0}^{T}e^{\left( T-s\right) A}A G \psi_1(s) ds= \int_{0}^{T}e^{\left( T-s\right) A}\left(\begin{array}[c]{cc}0 I -{{\Lambda}\,}0\end{array}\right) \left(\begin{array}[c]{c}0 \psi_1(s)\end{array}\right) ds
\limsup_{n\to\infty}\frac{1}{b_c^{(n)}}\log P(B_n^{(3)}) &\,\,\,\leq\limsup_{n\to\infty}\frac{1}{b_c^{(n)}}\log[(\lfloor n-\varepsilon f_2(n)\rfloor-\lfloor n-\varepsilon' f_2(n)\rfloor)P(S_n(t_n)\leq t_n-a_n)] &\,\,\,=-H(\ell_2\varepsilon),
a_{\delta}\geq \frac{1-c_4\delta^{\alpha \frac{2}{n+1}}(1+s_4(\delta'))}{1-c_2\delta^{\frac{2}{n+1}}(1+s_2(\delta))}=1+c_2\delta^{\frac{2}{n+1}}(1+o_{\delta}(1)+o_{\delta'}(1))\quad.
-{1\over 4\mu_0 l_\gamma}\int ds\sqrt{\overline y^{\prime\,2}(s)}{\delta^2 \Psi[\,\sigma^{\mu\nu}\ ; A]\over \delta\sigma^{\mu\nu}(s)\sigma_{\mu\nu}(s)}=i{\partial \over\partial A}\Psi[\,\sigma^{\mu\nu}\ ; A]
\begin{aligned}\dot{R}(t) =-\iint_{\mathbb T \times \mathbb R}\sin(\theta-\phi)\omega f(\theta,\omega,t) ~ d\theta d\omega &+KR(t) \iint_{\mathbb T \times \mathbb R}\sin^{2}(\theta-\phi)f(\theta,\omega,t) ~ d\theta d\omega.\end{aligned}
\mathbb{E}[{|X^* A X - \mathrm{Tr}(A)|}^p] \le 2^{p-1}(\mathbb{E}[|\sum_{i=1}^N (X_i^2 - \mathbb{E}[X_i^2])a_{ii}|^p] + \mathbb{E}[{|\sum_{i \neq j}X_i X_j a_{ij}|}^p]).
{\frac{1 - w}{\tan^{2}\psi_{1}}}={\frac{\tan^{2}\psi_{2}}{1 - x^{2}}}={\frac{1 - v}{\tan^{2}\psi_{3}}}={\frac{1}{\tan^{2}\delta}}= G : ={\frac{v w}{x^{2}}}- ( 1 - v ) ( 1 - w )
u=Q^{-1}(\sigma)r=\frac{y^{*}_{1}r}{\lambda_{1}-\sigma}\cdot x_{1}+\frac{y^{*}_{2}r}{\lambda_{2}-\sigma}\cdot x_{2}+\sum^{n}_{i=3}\frac{y^{*}_{i}r}{\lambda_{i}-\sigma}\cdot x_{i}.
\mathrm{Vol}(N) \leq \mathrm{Vol}\left(B^{M^k(-\lambda)}_{\mathrm{Diam}(N)}(x')\right) = \frac{2\pi^{\frac{k}{2}}}{\Gamma(\frac{k}{2})}\int_0^{\mathrm{Diam}(N)}\left(\frac{\sinh(t\sqrt{\lambda})}{\sqrt{\lambda}}\right)^{k-1}dt.