\frac{d}{d\epsilon} J_1(u_1 + \epsilon v_1, u_2) \Big|_{\epsilon=0}


\frac{d}{d\epsilon} J_1(u_1 + \epsilon v_1, u_2) \Big|_{\epsilon=0}=I_1 + I_2


-1<\beta_{\vert X \vert} \leq \ldots \leq \beta_2< \beta_1=1


R\Gamma_{\text{global}}(E, \mathcal{D}) \simeq R\text{Hom}(R\Gamma_{\text{global}}(E, \mathcal{D}), \mathbb{Q}/\mathbb{Z}(1))[1]


C^\bullet(E) \simeq R\text{Hom}(C^\bullet(E), \mathbb{Q}/\mathbb{Z}(1))[1]


\Lambda(E, s) = \varepsilon_E \cdot \Lambda(E, 2 - s)


R\Gamma(\mathbb{Q}_v, \mathcal{D}_v) \simeq R\text{Hom}(R\Gamma(\mathbb{Q}_v, \mathcal{D}_v), \mathbb{Q}/\mathbb{Z}(1))[1]


\vdash \phi \quad \text{iff} \quad M \vdash g \tag{\text{$\dagger$}}


h: \mathbb{R}^p \times \mathbb{R}^p \to \mathbb{R}


\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)


\log(z) = \log|z| + i(\arg(z) + 2\pi m), \forall m \in \mathbb{Z}


B_{kj} = \begin{cases} \delta_{k, j-1}, & \text{for } k = 1, 2, \ldots, 2r-1 ,\\ -\dfrac{h_{j-1}}{\tilde{a}_r - i \tilde{b}_r}, & \text{for } k = 2r, \end{cases}


x_t = (\arg(z_t) + 2\pi m) - i \log|z_t|, \quad t = 1, 2, \ldots, 2r, \quad \forall m \in \mathbb{Z}


h_j = \begin{cases} \tilde{a}_{r-j} + i \tilde{b}_{r-j}, & j = 0, 1, \ldots, r-1 ,\\ 2 \tilde{a}_0, & j = r, \\ \tilde{a}_{j-r} - i \tilde{b}_{j-r}, & j = r+1, r+2, \ldots, 2r. \end{cases}


\tilde{a}_k = \begin{cases} 0, & \text{for } k = 0\\ b_k k, & \text{for } k = 1, 2, \ldots, r \end{cases} \quad \text{and} \quad \tilde{b}_k = -a_k k, \text{for } k = 1, 2, \ldots, r


x_t = \arg(z_t) + 2\pi m, \quad \text{when} \quad |z_t| = 1


V(s) \approx \hat{v}_w(s) \coloneqq w^\intercal \psi(s)


S_F: U(b)\subset L^q([0,b], H_p)\multimap L^q([0,b],L^p(D)^3)


\left\{ \begin{aligned} \varphi(0,\mu) &= \mu^{2} + const, \quad \text{if } \mu > 0, \\ \varphi(1,\mu) &= \mu^{2} + const, \quad \text{if } \mu < 0. \end{aligned} \right.


\varphi(x,\mu) = \mu^{2}\cos^{4}\pi x + const


s(x,\mu) = -4 \pi \mu^{3} \cos^{3}\pi x \sin\pi x + \Sigma_{t}(\mu^{2}\cos^{4}\pi x + const) - \Sigma_{s}(const + \frac{\cos^{4}\pi x}{3})


|f|, |g|: |\mathcal{P}| \to |\mathcal{Q}|


\vec{a}_1 = (0, \frac{n}{2}), \quad \vec{a}_2 = (1, \gamma), \quad \text{with } 0 \leq \gamma < \frac{n}{2}


(1\otimes \Delta_w) \Delta_{z+w}=(\Delta_z\otimes 1)\Delta_w


{\mathrm{Span}\left(\Bigl\{ u_i \Bigr\}_{i=2}^n\right)=\mathrm{Span}\left(\Bigl\{ v_i^\ast \Bigr\}_{i=2}^n\right)}


\mu_u^\perp=\sum_{i=2}^n\,\left(u_i\otimes u_i^\dagger-u_i^\ast \otimes u_i^t\right)


\iota_1\circ c' = \iota_0\circ c\circ \alpha^{-1}


c'\circ\alpha = \overline{\alpha} \circ c


\cos(\theta_{12}) = \frac{1}{e} \left( \frac{p}{r^F} - 1 \right)


l^F_2 = \Omega^F + \omega^F + n^F (t_2 - t^F_p)


d_{\Sigma} \triangleq \max_{(d_1,\cdots, d_K)\in \mathcal{D}}(d_1+\cdots +d_K)


\overline{p(\Gamma)} = \R_{\bf H} \times \R_{\bf L}


\R_{\bf H} \times \R_{\bf L} \simeq \R^2


(\tilde \Omega, \tilde{\mathscr{F}}, \tilde{\mathbb{P}})


\sigma: [0, T]\times \mathbb{R}^d \times S \mapsto \mathbb{R}^{d \times \tilde{d}}


b:[0, T]\times \mathbb{R}^d \times S \mapsto \mathbb{R}^d


\tilde{\mathbb{F}}^r := \{\tilde{\mathscr F}_t^r\}_{t\geq 0}


\tilde{\mathscr F}_t:=\tilde{\mathscr F}_t^{B} \vee \tilde{\mathscr F}_t^r


\begin{aligned} &v^n\;m(u,v)\left(Q\left(\frac{1}{v},\frac{u}{v}\right)-uP\left(\frac{1}{v},\frac{u}{v}\right),-vP\left(\frac{1}{v},\frac{u}{v}\right)\right) \text{ in } U_1, \\ &v^n\;m(u,v)\left(P\left(\frac{u}{v},\frac{1}{v}\right)-uQ\left(\frac{u}{v},\frac{1}{v}\right),-vQ\left(\frac{u}{v},\frac{1}{v}\right)\right) \text{ in } U_2, \\ &m(u,v)(P(u,v),Q(u,v)) \text{ in } U_3, \end{aligned}


\lambda \in \left (\bigcup_{x\in \partial M} i |P_n\left [ -\sqrt{\max(0,N^2)},\sqrt{\max(0,N^2)}\right ]\right )^c


\nabla \cdot (\rho_0 z_v) + \frac{g_0'}{c^2} \cdot \nabla \times (\rho_0 w_v) + \frac{\rho_0 g_0'}{c^2} \cdot z_v = 0 , \quad n \cdot z_v|_{\partial M} = 0


w = \nabla \times (\rho_0 w_v) + \nabla \varphi_v


u = \nabla \times (\rho_0 w_u) + \rho_0 z_u


P_{XY}=\begin{pmatrix} 1-s & 0 \\ sd & s(1-d) \end{pmatrix}


H_N = \text{span\;}\{e_k : k = 1,\dots, N \}


\begin{aligned} &|\sigma(\mathbf{u})|^2=\sum_{k=1}^{m}|\sigma_k(\mathbf{u})|^2\leq B_0,\text{ for all }\mathbf{u}\in H;\\ &|\sigma(\mathbf{u})-\sigma(\mathbf{v})|^2=\sum_{k=1}^{m}|\sigma_k(\mathbf{u})-\sigma_k(\mathbf{v})|^2\leq L|\mathbf{u}-\mathbf{v}|^2,\text{ for all }\mathbf{u},\mathbf{v}\in H. \end{aligned}


|\sigma(\mathbf{u})^{-1}(P_N\mathbf{w})|\leq C_0|P_N\mathbf{w}|,\text{ for all }\mathbf{u},\mathbf{w}\in H


V := \{\mathbf{u}\in H^1(D)^2 :\nabla\cdot\mathbf{u}=0,\; \mathbf{u}|_{\partial D}=0\}


P_NH\subset \text{Range}\;(\sigma(\mathbf{u})),\text{ for all }\mathbf{u}\in H


c_{m,q}\neq c_{m',q}, \forall q, m\neq m'


\Omega = \prod_{i=1}^d (0,L_i) \subset \mathbb R^d


S^h = \{ \zeta \in C^0(\overline{\Omega}) \, : \, \zeta \vert_{o} \in P_1(o) \quad \forall o \in \mathcal{T}_h\}


\chi _{2k+1}(f)^2\leq \lambda _k(f)\lambda _{k+1}(f)


f\times f:X\times X\rightarrow X\times X


\max _{j=0,\ldots ,n}\lambda _j(Fr_q^s\circ f^t)


\Gamma(\Z)=\left\lbrace j \mid\|\Z(:,j,:)\| \neq 0, \, j=1, \ldots, r\right\rbrace


\Z \in \mathbb{R}^{n_2 \times r \times b}


\xi_n \to \xi_\infty \in [0,1]^N,\qquad x_n/t_n\to \zeta\in \partial\widetilde{W}


(0, \mu^{\sharp\prime}, \ldots, \mu^{\sharp\prime})


\omega_1^\sharp, \ldots, \omega_{n-2}^\sharp, \omega_{n-1}^\sharp+\omega_{n-2}^\sharp, 2\omega_{n-1}^\sharp, 2\omega_{n-2}^\sharp


\tilde{s}_{C, \epsilon}(p, q, r) = \begin{cases} -\frac{2}{q} + \epsilon, & \text{for } q \geq 3p' \text{ and } \frac{1}{p} \leq \frac{1}{4}; \\ \frac{2}{p} - \frac{2}{q} - \frac{1}{2} + \epsilon, & \text{for } q \geq 3p' \text{ and } \frac{1}{4} < \frac{1}{p} < \frac{1}{2} - \frac{1}{q}; \\ \frac{1}{p} - \frac{3}{q} + \epsilon, & \text{for } q \geq 3p' \text{ and } \frac{1}{p} \geq \frac{1}{2} - \frac{1}{q}; \\ \frac{3}{2p} - \frac{3}{2q} - \frac{1}{2} + \epsilon, & \text{for } p' < q < 3p'; \\ \frac{2}{p} - \frac{1}{q} - 1 + \epsilon, & \text{for } q \leq p'. \end{cases}


\theta_0 = \arctan(\sqrt{\frac{\phi_0}{\phi_2}} \phi_1)


\begin{aligned} \frac{\partial x}{\partial \xi} = a_0\xi + a_1\eta + a_2, \\ \frac{\partial x}{\partial \eta} = a_3\xi + a_4\eta + a_5, \\ \frac{\partial y}{\partial \xi} = b_0\xi + b_1\eta + b_2, \\ \frac{\partial y}{\partial \eta} = b_3\xi + b_4\eta + b_5, \\ \end{aligned}


I_n = \int \sec^n \theta \rm{d} \theta = \frac{1}{n - 1} (tan\theta \sec^{n - 2} \theta + (n - 2)I_{n-2}). \nonumber


\int \tan^{2k+1} \theta \sec^3 \theta \rm{d} \theta = \int (sec^2 \theta - 1)^k sec^2 \theta \rm{d} \sec \theta


\int \tan^{2k} \theta \sec^3 \theta \rm{d} \theta = \int (sec^2 \theta - 1)^k sec^2 \theta \rm{d} \theta


\min_{T} \|\mathbf{X} T - \mathbf{Y}\|^2


C_T \left(u^T_n X^{\text{Test}}_n,\overline{ \text{x}}^{\text{Test}}\right)


C_L \left(u^L_n X^{ \text{Work}}_n,0\right)


X^\mathbb{G}:= X^\odot \cup \bigcup\limits_{x \in X} \mathrm{L}_\mathbb{G}(x)


x\in Y\cap \bigcap_{J\in \mathcal{J}}N(J)


B(\bm z; r) = \{\bm x \mid |\bm x - \bm z| \le r\}


-\Delta g = \eta \quad\text{in }\; \Omega, \qquad g = 0 \quad\text{on }\; \Gamma


A(\bm z; r, R) = \{\bm x \mid r \le |\bm x - \bm z|\le R\}


\emptyset \neq T \cap \Gamma_h \subset S


\text{GELU}(x) = x \cdot \frac{1}{2}[1+erf(\frac{x}{\sqrt{2}})]


\langle L_1 \phi , \chi_b(r) r^{\frac12}\rho_1\rangle = \lambda_0 \langle \phi, \chi_b(r) r^{\frac12}\rho_1\rangle


\begin{split} \big(D^X(s-)+\tfrac12 \gamma(s) \Delta X(s)\big)^\top \Delta X(s) & = D^X(s-)\Delta X(s-) + \tfrac12 (\Delta X(s))^\top \gamma(s) \Delta X(s) . \end{split}


D^X(s)=D^X(s-)+\Delta D^X(s) = D^X(s-)+ \gamma(s)\Delta X(s)


g_{u}(u) := \rho \left[ \frac{u}{\rho}\right]


E_j(\theta) = E_{\Gamma_0}(\theta) + \mu_j


E_{\Gamma_0}(\theta) = 2\cos 2\pi\theta_1+2\cos 2\pi\theta_2+2\cos 2\pi(\theta_1+\theta_2)


E_j(\theta) = (1+\mu_j) E_{\Gamma_0}(\theta)+\mu_j


\langle a\rangle_p = \sum_{q=1}^{\nu_F} \langle a(\cdot+v_q)\rangle \sum_{s=1}^{\nu'_F} |P_{\mu_s}(v_p,v_q)|^2\,


E_j(\theta) = \mu_j E_{\Gamma_0}(\theta)


P_{\mu_s}(v_p,v_q) = \sum_{j\,\mu_j=\mu_s} w_j(v_p)\overline{w_j(v_q)}


|\langle L_I,h\rangle| \leq \alpha \int_0^T \int_{\mathbb{R}^N} |\xi(t,x)|dx dt + 2\beta \|I\|_{L^\infty}\|h\|_{L^\infty}T + \gamma \int_{\mathbb{R}^N} |\xi(T,x)|dx \leq C_1 \|h\|_{L^\infty(0,T)}


\Phi_I(\varepsilon)= \dfrac{J(I^\varepsilon) - J(I)}{\varepsilon}


\lim_{\varepsilon \to 0} \left\|\frac{G(I + \varepsilon h) - G(I)}{\varepsilon} - (\xi, \eta)\right\|_X = \lim_{\varepsilon \to 0} \left\|(\tilde{\xi}, \tilde{\eta})\right\|_X = 0


(p^\varepsilon, d^\varepsilon) = G(I^\varepsilon)


x.\omega=\omega(S(x).-),\; \forall\omega\in V^\ast, x\in A


\Phi(t, \pm 1) = \partial_y \Phi(t,\pm 1) = 0


H \coloneqq (\delta_{m+i, j})_{i=1, \dots, d-m; \: j=1, \dots, d}


g_k := \log\left(\frac{\beta}{\alpha}\right) - \frac{n}{2}\left(\frac{1}{\alpha^2}-\frac{1}{\beta^2}\right)(X_{k/n}-X_{(k-1)/n})^2


\mathsf{Diam}(H)=\max_{u,v \in V(H)} \mathsf{Dist}_H(u,v)


|N_{G}(U\setminus B)|\geq \frac{s|U\setminus B|}{2r}


|U'|\leq \frac{r|X \cap L|}{t}\le \frac{r}{t}\cdot |L| \leq \frac{r}{t}\cdot \frac{|U|\cdot t}{10r}= \frac{|U|}{10}


|L| = |C|\cdot t \le \frac{|U|}{10r}\cdot t


\beta (r\wedge s)^{-\gamma} (r\vee s)^{\gamma-1}


-\sum_{k=r}^m \frac1k < V(v) \leq -\sum_{k=r+1}^m \frac1k


(\theta_{P},\lambda_{P}) := \text{arg} \left( \max_{\lambda} \min_{\theta} L_{P,M}(\theta,\lambda) \right) \quad \text{resp.} \quad (\theta_{R},\lambda_{R}) := \text{arg} \left( \max_{\lambda} \min_{\theta} L_{R,M}(\theta,\lambda) \right)


L_{P,M}(\theta,\lambda) := J_{P,M}(\theta)+\frac{1}{X(\partial \Omega)} \sum_{{\bf x} \in X(\partial \Omega)} (U({\bf x};\theta)-g({\bf x}))\lambda({\bf x})


J_{R_I}(\theta) := \int_{\Omega} \left( \frac{1}{2} |\nabla \widetilde{U}({\bf x};\theta)|^2- f({\bf x}) \widetilde{U}({\bf x};\theta)\right) \, d{\bf x}


L_{R,M}(\theta,\lambda) := J_{R,M}(\theta)+\frac{1}{X(\partial \Omega)} \sum_{{\bf x} \in X(\partial \Omega)} (U({\bf x};\theta)-g({\bf x}))\lambda({\bf x})


\lambda=\lambda+ \alpha \nabla_{\lambda} L_{P,M} \quad \text{or} \quad \lambda=\lambda+\alpha \nabla_{\lambda} L_{R,M}


\mathcal{C}(2k)\simeq \mathbb{C}(2^k) \qquad \text{and}\qquad \mathcal{C}(2k+1)\simeq \mathbb{C}(2^k)\oplus\mathbb{C}(2^k)


s_{r,s}=\begin{cases} 2 \text{ if } r-s=2,4 \text{ mod 4},\\ 1 \text{ if } r-s=1,3 \text{ mod 4}. \end{cases}


\nabla_{\partial_{t_j}} \left( L(\tau,z) \alpha \right) =0


\mu(\phi_\ell)=\frac{1}{2} (\deg \phi_\ell - \dim X) \phi_\ell


(\tau,z)=(\sum_{j=1}^{b^{2}(X)} t_j \phi_j, z) \in H^2(X) \times \mathbb{C}^\times


\langle \alpha , \beta , \gamma \rangle_{0,3,d} = 0


\deg \alpha + \deg \beta + \deg \gamma \neq \langle c_1,d \rangle


\left\{ \left\Vert \nabla u_{n}\right\Vert _{L^{m}}:p_{n}^{-}\geqslant m\right\}


\left( W\left( x+x_{0}\right) -W\left( x_{0}\right) \right) _{x\in \mathbf{R}^{d}}


\left( W\left( x\right) \right) _{x\in \mathbf{R}^{d}}


\begin{aligned} (x;q)_\infty=\prod_{k=0}^\infty (1-xq^k),\qquad (x;q)_n=\prod_{k=0}^{n-1}(1-xq^k),\qquad (x;q)_0=1, \end{aligned}


1.262 \cdot \left(\lvert e \rvert\right)^{\frac{1}{5}} + 2


t_1\coloneqq\sup\{t\ge0\ |\ d_X(o,\gamma(t))\le R\}


\begin{split} d_X(x_n,p)&\le d_X(x_n,p_n)+d_X(p_n,p)\\ &=d_X(o,p_n)-d_X(o,x_n)+ d_X(p_n,p)\\ &=d_X(o,p_n)-R+d_X(p_n,p)\xrightarrow{n\to \infty} d_X(o,p)-R+0=0. \end{split}


(\partial_P X_n,\rho_{x_n})\xrightarrow{AI conv.}(\partial_P X,\rho_x)


\ker \tilde F_n\subset \ker \tilde F_{n-1}


\hat{G}^{(1)}=\{\xi \in \hat{G}\mid L_{\xi} \text{ has a single component}\}


\mathbb{E}[\|\hat{x}_\eta-x^*\|^2]\leq \tfrac{2}{\mu_f}\left(\|\nabla f(x^*)\| \,\sqrt[\kappa]{\tfrac{1}{\alpha}\left(\texttt{Err}_{\eta}+\eta M\right)} + \tfrac{1}{\eta}\texttt{Err}_{\eta}\right)


\tfrac{\mu_f}{2}\mathbb{E}[\|\hat{x}_\eta-x^*\|^2]+\nabla f(x^*)^\top\mathbb{E}\left[\left(\hat{x}_\eta- \Pi_{X^*_h}[\hat{x}_\eta]\right)\right]\leq \mathbb{E}[f(\hat{x}_\eta)] - f^*


0\le \mathbb{E}[h(\hat{x}_\eta)]-h^*\le \texttt{Err}_{\eta}+\eta M


0\le \mathbb{E}[ h(\hat{x}_\eta)]-h^*\le \texttt{Err}_{\eta}+\eta M


\mathbb{E}[h(\hat{x}_\eta) + \eta f(\hat{x}_\eta)] -f^*_\eta \leq \texttt{Err}_{\eta}


\psi(\omega_0) \coloneqq \log b(\omega_0)


\phi(\omega_0) \coloneqq - \log a(\omega_0)


\int \left| \log \sin \angle (\mathbf{E}_1, \mathbf{E}_2) \right| \, d\mu


F(\omega) = \begin{pmatrix} a(\omega_0) & b(\omega_0) \\ 0 & 1 \end{pmatrix} \,


\lbrace \mathbf{E}_1(\omega), \mathbf{E}_2(\omega)\rbrace = S


-K_X\cdot D = f_t^*\left(\sum_i C_i\right)\cdot D_t =\left( (p+q)E_t + \sum_i C_{i,t}\right) \cdot D_t = p+q


-2\leq (K_{Y_t} + D_t) \cdot D_t = \left(-\sum_{i}C_{i,t} - E_t + D_t\right) \cdot D_t = D_t^2 - E_t\cdot D_t < -E_t\cdot D_t


p+q = D\cdot C \geq (D\cdot C)_x = \alpha + \beta \geq p + q


f_t^*(\sum_{i}C_i)\cdot D_t = -K_X \cdot D> 0


\begin{aligned} & \underset{\mathbf{b}}{\text{minimize}} & & \epsilon_d(\mathbf{b}) = \dfrac{1}{n_{MP}} \sum_{i=1}^{n_{MP}} (\mathbf{u}_{r_i} - \mathbf{u}_{s_i}(\mathbf{b}))^2\\ & \text{subject to} & & \mathbf{b}_l \mathbf{\leq b} \leq \mathbf{b}_u \end{aligned}


\Pr{g(X) \geq k \text{ or } g(-X) \geq k} \leq 2\epsilon


\Pr{g(-X) \geq k} = \Pr{g(X) \geq k} \leq \epsilon


\langle Au,u-v\rangle\le\Phi(v)-\Phi(u)+I_\Delta(\psi^0(\gamma_\psi u;\gamma_\psi v-\gamma_\psi u))-\langle f,v-u\rangle


\langle Au^h, u^h-v^h\rangle \le \Phi(v^h)-\Phi(u^h) + I_\Delta(\psi^0(\gamma_\psi u^h;\gamma_\psi v^h-\gamma_\psi u^h))-\langle f,v^h-u^h\rangle


\int_{-\infty}^t F_X(x)dx\geq \int_{-\infty}^t F_Y(x)dx\ \text{ for all }t


\int_t^{\infty} \bar F_X(x)dx\leq \int_t^{\infty} \bar F_Y(x)dx\ \text{ for all }t


E^{2t} \cap \{B_u = \tilde{B}_u, \forall u \in \partial G_e^{2t}\}


E^{2t+1} \cap E^{2t} \cap \{B_u = \tilde{B}_u, \forall u \in \partial G_e^{2t}\}


u \in \partial G_e^{2t}\equiv \partial T_e^{2t}


|\partial G_e^{2t}| \le (2k\lambda+2)^t \log n


\omega_1 = \frac{\cos^2\theta}{a^2} + \frac{\sin^2\theta}{b^2}, \qquad \omega_2 = \frac{\cos^2\theta}{b^2} + \frac{\sin^2\theta}{a^2}, \qquad \omega_3 = \sin2\theta(\frac{1}{a^2}-\frac{1}{b^2})


\alpha'=\widehat\beta,\quad \beta'=\widehat\gamma,\quad {\rm and}\quad \delta'= \widehat\alpha~


S_z(\omega) = \frac{1}{N} \sum_{\ell=1}^N S_{x_\ell}(\omega)


\left\langle e^{i(\theta_m(t') - \theta_n(t))} \right\rangle = \delta_{mn} \left\langle e^{i(\theta_m(t') - \theta_m(t))} \right\rangle


S_{x_\ell}(\omega) = \lim_{T \to \infty} \frac{1}{T} \left\langle |\tilde{x}_\ell(\omega)|^2 \right\rangle, \quad \tilde{x}_\ell(\omega) = \int_0^T e^{i\omega t} x_\ell(t)\,dt


m \ddot{\theta}_\ell(t) + \dot{\theta}_\ell(t) = \omega_\ell + \text{Im}\left( e^{-i\theta_\ell(t)} \zeta_\ell(t) \right)


S_z(\omega) = \frac{1}{N} \sum_{\ell=1}^{N} |\tilde{x}_\ell(\omega)|^2


\tilde{\varepsilon}^i \sim \mathcal{N}(0, \pmb{I}_l)


\epsilon_{\text{Darcy}} \sim N(0, \pmb{I})


\Delta{t}_{\text{Filter}} = T/ Nt_{\text{Filter}}


\Theta_*=\{\phi,\dot{\phi}, \cos \phi,\sin \phi,\phi^2,\phi \dot{\phi},\ldots \}


\langle \tau, \omega\rangle =\frac{2-\|\tau-\omega\|^2}{2}, \quad \forall \tau, \omega \in \mathbb{R}^d


(q^n_m(e^n_i, z_1, \ldots, z_n), q^n_m(e^n_j, z_1, \ldots, z_n)) \in \theta


(b_1, \ldots, b_n, 0, \ldots, 0) \in T(m)


B \simeq B / \theta(c,1) \times B/\theta(c,0)


\alpha_{n_0}\le \alpha_{n_1}\le \cdots\le \alpha_{n_k}


\lim_{n \to \infty} P_n\left(\sup_{t \geq 0}\left|\frac{1}{\sqrt{n}}\sum_{k \leq t\sqrt{n}}X_k- L(t)\right|\leq n^{-\frac{1}{4}+\delta} \right)=1


\limsup_{n \to \infty} n^{-\delta}\log P_n\left(\inf_{t \geq 0}\left|\frac{1}{\sqrt{n}}\sum_{k \leq t\sqrt{n}}X_k- L(t)\right|> n^{-\frac{1}{4}+\delta} \right) <0


\phi(\mu,E,\hbar ,x) = \sum_{k=0}^\infty \phi_k(\mu ,E,x) \hbar^k


\tilde{\phi}_g: \mathcal T_c(E_{g^{-1}}) \rightarrow \mathcal T_c(E_g)


E_g = \{x \in E \colon x \leq ss^{\ast} \text{ for some } s \in S \text{ with } \varphi(s) = g\} \subseteq E


(\mathcal T_c(E_1), \Phi_1) \subseteq (\mathcal T_c(E_2), \Phi_2)


\Vert \delta H \Vert < \mu_{(0,0,E)}(X,Y,H)


v(0,\cdot) =v_{\epsilon}^1(0,\cdot) + \varphi_0, \quad \partial_t v(0,\cdot) = \partial_t v_{\epsilon}^1(0,\cdot) + \varphi_1


\|\varphi (-\log(T-t) + \log T,\cdot) \|_{\dot H^{r}(\R^n)} \lesssim \delta (T-t)^{\omega}


\|(\partial_0 + \Lambda + 1) \varphi(-\log(T-t) + \log T,\cdot) \|_{\dot H^{r-1}(\R^n)} \lesssim \delta (T-t)^{\omega}


v(t,x) = v_{\epsilon}^{T}(t,x) + \frac{1}{T-t} \varphi \left (\log\left(\frac{T}{T-t}\right), \frac{x}{T-t} \right)


T = T_{\epsilon} \in [1-\delta , 1+\delta]


A:= \begin{bmatrix} -\frac{x_1}{\Theta_1}\alpha_1 & \alpha_1 \\ -\beta_1 & 0 \end{bmatrix} \begin{bmatrix} -\frac{x_0}{\Theta_0}\alpha_0 & \alpha_0 \\ -\beta_0 & 0 \end{bmatrix} = \begin{bmatrix} \frac{x_0x_1}{\Theta_0\Theta_1}\alpha_0\alpha_1-\beta_0\alpha_1 & -\frac{x_1}{\Theta_1}\alpha_0\alpha_1 \\ \frac{x_0}{\Theta_0}\alpha_0\beta_1 & -\alpha_0\beta_1 \end{bmatrix}


\ddot{\kappa}(t)^{\top} = \frac{d}{dt} \left[ D_2 C(q(t),\dot{q}(t),u(t)) - \kappa(t)^{\top} D_2 f(q(t),\dot{q}(t),u(t)) \right]


T^{(2)} \mathcal{Q} \oplus_{T\mathcal{Q}} \mathcal{E}


\tilde{K}_{k\Delta t, U(\cdot, 0), \theta}\coloneqq \max_{1\leq i\leq k}K_{i\Delta t, U(\cdot, 0), \theta}


\begin{aligned} W_{2, \delta}^{2, \text{e}}\big(\hat{U}(\cdot, k\Delta t;\hat{\theta}), \hat{U}_n(\cdot, k\Delta t;\hat{\theta} )\big) &\leq \sup_{\hat{\theta}, U(\bm{x}, 0)} K_{T, U(\cdot, 0), \theta}\lambda_{N+1}^{-1}(\hat{\theta}). \end{aligned}


\big(U(\bm{x}, k\Delta t; \theta), U_n(\bm{x}, k\Delta t;\theta)\big)


\lambda_{-1}(\omega) \le \lambda_{-1}(0) + \lambda^{\prime}_{-1}(0) \omega


\cdots 211 211 \cdots 211 \bullet 2(1)^{\infty}


\cdots 210 210 \cdots 210 \bullet 2 (1)^{\infty}


d(l_{\beta}, \beta) = \bullet2 (1)^{\infty}


\iota:Q_{\bf d}\rightarrow Q_{\bf e},\;\;\; (i,k)\mapsto(i,k+e_i-d_i)


R_{\widehat{\bf d}}^{\rm flag}(Q_{\bf d})


\iota:W_{Q_{\bf d}}\rightarrow W_{Q_{\bf e}}


A_1 f= \nabla f \text{ and } A_2 \omega = F \cdot \omega


{A_F}_{\alpha,\beta} =\langle A_F e_\beta, e_\alpha\rangle


\left(A_1 f\right)_l = \sum_{\beta \in \mathbb N^n} (\beta_l + 1) f_{(\beta_1, \ldots, \beta_{l-1}, \beta_l + 1, \beta_{l+1},\ldots, \beta_n)} e_\beta


A_F f = \sum_{\alpha\in \mathbb N^n} \left( \sum_{\beta\in\mathbb N^n} {A_F}_{\alpha,\beta} f_\beta \right)e_\alpha\\


|H'_{y'}| \geq c(1 + |\theta'|), \quad \text{for } |\theta'| \geq C > 0, \text{ large}


Q(a e^{i \frac{\varphi}{h}}) = Q'_\varphi({a}) e^{i \frac{\varphi}{h}} + R(a), \tag{3.12}


\varphi(x, \xi') - \varphi(y, \xi') = \Sigma(x, y', \xi')(x' - y')


x_{k+1} = x_k - \Gamma \nabla_{x_k} V(x_k)


\varphi_\Theta : [0, +\infty) \to [0, +\infty)


\varphi_{p,\Theta} : [0, +\infty) \to [0, +\infty)


\Theta_n : [0, +\infty) \to [0, +\infty)


V\cong\operatorname{Im}\pi \oplus \ker\pi=W\oplus W'


f_k := (q^k;q^k)_{\infty}, \quad \text{for any integer} \quad k\geq1


p^*i_0^*(i_1^*b_J + \delta \phi) = \delta v


\mathbb{F} = (\mathcal{F}_t)_{t \geq t_0}


\mathcal{V}:=\{(v,q)\in H^1_0(\Omega)\times L^2_0(\Omega)\colon -\Delta v+\nabla q \in L^2(\Omega), \nabla\cdot v\in P\}


u\in H^s(\Omega),\quad p\in H^{s-1}(\Omega) \quad\forall s<1+\lambda


\dfrac{|\nabla w_1(x)|}{w_1(x)}=\dfrac{|\nabla w_2(x)|}{w_2(x)}\ \Longrightarrow\ \dfrac{|\nabla w_1(x)|}{w_1(x)}=\dfrac{\lambda(x)|\nabla w_1(x)|}{w_2(x)}\ \Longrightarrow\ \lambda(x)=\dfrac{w_2(x)}{w_1(x)}


|\nabla w_1(x)|\cdot |\nabla w_2(x)|=\nabla w_1(x)\cdot\nabla w_2(x)


H^* = \operatorname{Span}\{f_1, f_2, \cdots, f_n\}


\{b_i\}_{i\leq n} \subseteq X_{\leq 1+\sigma}


(\lambda_1, \lambda_2, \cdots, \lambda_n)\in\mathbb{C}^n


\rho_n = \sup\big\{ r(y)\,\vert\, y\in X^{\ast}_{= 1}\cap E_n^{\perp}\big\}


p(\tau, \tau \sqcup \{ij\}) = \left\{ \begin{array}{cl} \displaystyle \frac{\alpha_{ij} w_{i, ij}}{\sum_{k=1}^d \alpha_{ik} w_{i, ik}} \alpha_i , & \text{if } i \in \partial \tau, \\[15pt] \alpha_{ij}, & \text{if } i \notin \partial \tau \text{ and } ij \notin \tau, \end{array} \right.


\alpha_i = \alpha_{i1} + \ldots + \alpha_{id} \in [0, 1]


{\prod_{i \in \overset{\circ}{\tau}} \frac{\alpha_i}{\sum_{j=1}^d \alpha_{ij} w_{i, ij}}}=\left(\frac{d}{\sum_{j=1}^dw_j}\right)^{|\tau|- |\partial \tau|}


\sigma^{(j)} \sqcup \bigsqcup_{l\neq j} \sigma^{(l)}


\mathcal M_{\varepsilon} \rightarrow e(F,\rho)


I_i = \frac 43 \int \rho_i^{\frac 32} dt


\rho(t)= \begin{cases} \frac{1}{a^2(3-\sqrt{8a})^2}, & \text{if } t \leq 2a \\ \frac{2}{at(3-\sqrt{8a})^2}, & \text{if } 2a \leq t \leq 1. \end{cases}


G(U)=\begin{pmatrix}g_{1}(U) \\ g_{2}(U) \\ g_{3}(U) \end{pmatrix}=\begin{pmatrix} m P(H(M)-a_{1} \lambda M ) \\ -m a_{2} \lambda M S \\m_{s}S(1-M)-\eta M P \end{pmatrix},\quad U_{R}=(0,S_{R},M_{R})\quad \text{and}\quad U_{0}=(P_0,S_{0},M_{0})


W^{2,1}(\Omega_R) = \{u \in L^1(\Omega_R) : D^\alpha u \in L^1(\Omega_R) \quad\text{for all}\quad |\alpha| \leq 2\}


\|f\|_{L^{\infty}(\Omega_R)} = \operatorname{ess\,sup}_{x \in \Omega_R} |f(x)|


\displaystyle \mathcal{A}\left(U\right)(t,x)=\left[\operatorname{div}\left(\vec{\alpha_{p}}(t,x)\left(\int_{ \Omega_{R}}\gamma_{p}(x-y)P(t,y)d y \right) P(t,x)\right),\operatorname{div}\left(\vec{\alpha_{s}}(t,x)\left(\int_{ \Omega_{R}}\gamma_{s}(x-y)S(t,y)d y \right)S(t,x)\right),-D\Delta M\right]


\int_H \phi(xh)dh = \int_{H^{\perp}} \mathcal{F}_A(\phi)(\chi)\chi(x)d\chi


\left( \alpha ,\varepsilon ,\lambda \right)


\mathcal{U}_T : f \in L^2(\R^n) \mapsto u(T, \centerdot) \in L^2(\R^n)


C_T = \frac{\partial \mathbb{E}[E(T)] }{\partial T}


t^* = \underset{t \geq M}{\arg\max} \, \hat{C}(t)


\sum_{i \neq j \in [n] : |i-j|\leq 2}\frac{1}{(|\lambda_{j}-\lambda_{i}| + n^{-1.5-\tilde{\epsilon}})^2} \leq 8c_0 n^{3.5 + 2\tilde{\epsilon}}


\min_{i \in [n-1]} |\lambda_{i+1}-\lambda_i| \geq n^{-1.5-\tilde{\epsilon}}


\min_{i \in [n-3]} |\lambda_{i+3}-\lambda_i| \leq n^{-6/5-\tilde{\epsilon}/5}


\sum_{i, j \in [n]: i \neq j} \frac{1}{(\lambda_j-\lambda_i)^2} \leq 32c_0 n^{3.5 + 2\tilde{\epsilon}} + \pi^2 n^{17/5+2\tilde{\epsilon}/5} \leq n^{3.5 + 3\tilde{\epsilon}}


\nabla\cdot(\bm{u}\otimes \bm{u})=(\bm{u}\cdot \nabla)\bm{u}+(\nabla\cdot\bm{u})\bm{u}=(\bm{u}\cdot \nabla)\bm{u}


(\bm{u}\otimes\bm{v})_{ij}=u_i v_j, ~(\bm{C}\otimes\bm{u})_{ijk}= C_{ij}u_k


\nu = \Omega\left(\frac{\omega}{1 + (\omega L)^2}\right)


\argmin_{x\in\mathbb{R}} f(x) = \frac{\eta}{\sigma^2+\eta^2} = \frac{\eta}{|z|^2}=:x_0


P \left( \sigma(W) \geq \frac{1}{k} \right)>0


P \left( N=n_0 \text{ for the sequence generated by }\{X_i\}_{i> n_0}\right) > 0


\frac{A(n)}{n} \geq \begin{cases} 1 & \text{if } n \leq n_0 \\ \frac{1}{2} & \text{if } n_0 < n \leq 2n_0 \\ \frac{1}{k} & \text{if } n > 2n_0 \end{cases}


N = \min\left\{n:\forall m \geq n, \quad \frac{A(m)}{m} > \frac{1}{k}\right\}


\begin{split} a(b+c) & = ab + ac \\ (a+b)c & = ac + bc \end{split}


L=\left(\begin{array}{cccc} 3 & 1 & & \\ 1 & \ddots & \ddots & \\ & \ddots & \ddots & 1 \\ & & 1 & 3 \end{array}\right)


\begin{aligned}\|\mu(n_1)-\mu(n_2)\|_X&\leq 2T\|\partial_Ir\|_\infty \|I_1-I_2\|_\infty. \\ &\leq 2T\|\partial_Ir\|_\infty r_M\|n_1-n_2\|_X \end{aligned}


\forall n \in X, \qquad \Phi(n,F(n)) = 0


\|\partial_nF(n)\|\leq \big\|\big(\partial_I\Phi(n,I)\big)^{-1}\big\|\:\big\|\partial_n\Phi(n,I)\big\|\leq 1\times r_M


\boldsymbol{h}, \boldsymbol{x} \in \mathbb{C}^n


D_p \in \mathbb{M}_{n\times n}(\mathbb{Q}_p)


|x|_p = p^{-k}, \quad \text{where } k \in \mathbb{N} \text{ and } k \geq 1


(D_p x)_i = \sum_{j=1}^n d_{ij} x_j \in \mathbb{Q}_p


\delta_p(x) = |D_p x|_p \in \mathbb{Q}_p^n


x, \delta_p(x), \delta_p^2= \delta_p(\delta_p(x))= \delta_p \circ\delta_p, \dots, \delta_p^n(x), \dots


\partial_{\lambda_{i}} \Phi = \sum_{j=1}^n l_{ij} - 1


\Phi(L,\lambda):=D(L,M) +\sum_{i=1}^n \lambda_i \left(\sum_{j=1}^n l_{ij} - 1\right)


\argmin_{L\in C_2} D(L,M) = \left(\frac{m_{ij}}{\sum_{k=1}^n m_{kj}}\right)_{1\le i, j\le n}


\argmin_{L\in C_1} D(L,M) = \left(\frac{m_{ij}}{\sum_{k=1}^n m_{ik}}\right)_{1\le i, j\le n}


\lim_{k\to\infty} M^k=\argmin_{L\in C} D(L,M^0) =\argmin_{L\in C} H(L)


\partial_{l_{ij}} \Phi = \log(l_{ij})-\log(m_{ij}) +\lambda_i


\partial_{x}G(t,s)=-\sum_{i=1}^n a_ib_i\exp\{-b_i(t-s)\}, \quad 0 \le s < t \le T


\frac{1}{2}\iota_{[\xi,\xi]}\delta_\chi\omega + \iota_\xi\mathrm{L}^\omega_\xi \delta_\chi\omega - \frac{1}{2}\iota_\xi\iota_\xi d_\omega \delta_\chi\omega=0


\max_{j} \left[ \bar{g}_z^j\left(h(z)\right) \right]


\beta_{\ell} = \prod_{r=2}^{\ell} \alpha_r


\theta_j = \delta \{ \varphi_j \}/2 = (\varphi_{j+1} - \varphi_j)/2


\lambda_j = -\partial_1 S(\varphi_j,\varphi_{j+1}) = \partial_2 S(\varphi_{j-1},\varphi_j)


2q \mu\big \{ h(\psi_j) \sin \theta_j \big \} = A\{ \varphi_j\} = L,\quad \mu \{ h'(\psi_j) \sin \theta_j \} = 0, \quad \mu \{\theta_j\} = \pi p/q


S(\varphi_j,\varphi_{j+1}) = 2 h(\psi_j) \sin \theta_j


\mathcal L(G^k_1)\sim \Gamma\left(\lambda,\frac{(k+c_+)(k+c_-)}{ 2 \sigma^2}\right)


\mathcal L(T_1) \sim \Gamma(\lambda, \theta_+\theta_-/2)


\forall i \in [k]\colon (v_i, v_{i+1}) \in E \lor (v_{i+1}, v_i) \in E


\forall i \in [k]\colon (v_i, v_{i+1}) \in E


N_j(G^{\sigma}[H^{\sigma}]) = \{ x \in V(G^{\sigma}) \setminus H^{\sigma} \mid d(x, G^{\sigma}[H^{\sigma}]) = j, \quad j = 1, 2, \dots, n \}


\left\lfloor \frac{g_r}{2} \right\rfloor + k \geq g_r


\left\lceil \frac{g_r}{2} \right\rceil \leq k


(V(P^\sigma) \setminus \{y, y'\}) \cap V(C^\sigma) = \emptyset


s_{ij} \,\,= \,\,\lambda_i \lambda_j \! \left( 1 - \langle x^{(i)}, x^{(j)} \rangle \right)


\ln \lambda_{s,E_{2}} - \ln \lambda_{s, E_1}


\{\Lambda(\omega_{\mid K}) : \omega \text{ smooth} \}


i=\underset{i\in T_{(l)}^{c}}{\arg\max}\underset{k}{\min}\ \|\mathbf{h}_k\|_2,\ \text{s.t.}\ \mathbf{h}_k\in\text{Col}(\mathbf{H}_{T_{(l)}\cup\{i\}})


\mathbf{G}, N_r, N_{\text{init}}, \Delta n


i=\underset{i\in T_{(l)}^{c}}{\arg\max}\ \bar{\mathbf{g}}_i,\ \operatorname{s.t.}\ D_1(\{i\},T_{(l)})\leq \Delta n


\alpha_{k}^{RBB}=\mathop{\text{argmin}}_{\alpha\in\mathbb{R}} \Big\{\Vert\alpha\Delta y^{k-1}-\Delta\Psi^{k-1}\Vert_{2}^{2} + \tau_{k}\Vert\alpha \sqrt{H_{k}}\Delta y^{k-1}- \sqrt{H_{k}}\Delta\Psi^{k-1}\Vert_{2}^{2}\Big\}


d_{\lambda} \coloneqq \mathrm{dim} \left(V_{\pi_{\lambda}}\right)


\hat{f}(\lambda)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{-i \lambda \phi} f(e^{i\phi}) d \phi \mathrm{,}


N_i+\left(\bigcap_{j \neq i} N_j\right)=M


M / \bigcap_{i \in I} N_i \rightarrow \oplus_{i \in I} M / N_i


M=N_{k}\oplus P_{k}\oplus P_{k-1}\oplus \cdots \oplus P_{1}\quad \text{with}\quad N_{k}=N_{k+1}\oplus P_{k+1}


C(X,Y,Z):=C_{ijk}X^iY^jZ^k=\frac{1}{4}\frac{\partial^3F^2(x,y)}{\partial y^i\partial y^j\partial y^k}X^iY^jZ^k


\frac{\delta}{\delta x^i}=\frac{\partial}{\partial x^i}-N^j_i\frac{\partial}{\partial y^j}


\sum_{k=0}^\infty d_{n,k}c_k = \sum_{k=0}^nd_{n,k}c_k = [t^n]d(t)c(h(t))\nonumber


\mathcal{F}(t) \coloneqq \overline{\sigma}(B(s), 0 \leq s \le t)


\mathbb{F} \coloneqq \{\mathcal{F}(t)\}_{t \ge 0}


\lambda^\sharp_{1}(k,T)=\sum_{c \leq T} (\tfrac{4n_1n_2}{c})


S_h = (\alpha_1, \alpha_2, \ldots, \alpha_r, \overline{\alpha}_1, \overline{\alpha}_2, \ldots, \overline{\alpha}_r)


S_h = \Phi_{\alpha h} \, \Phi_{\overline{\alpha} h}


T(u)\equiv f(u)- P_{\Phi(u)}(f(u)-\alpha u)


\dot{V}(u) \leq -K.\big(V(u)\big)^p \quad \forall u\in U\setminus \{u^*\}


N = \left\lfloor \frac{1}{\gamma} \right\rfloor


\overbrace{ \begin{pmatrix} \begin{matrix}a_m&\ldots&a_0\\ &\ddots&&\ddots& \\ &&a_m&\ldots&a_0\\\end{matrix} \\ \begin{matrix}b_n&\ldots&b_0 \\ &\ddots&&\ddots& \\ &&b_n&\ldots&b_0\\\end{matrix} \\ \end{pmatrix}}^{n+m-i} \begin{matrix} \left.\begin{matrix}\\ \\ \\\end{matrix}\right\}&n-i\\ \left.\begin{matrix}\\ \\ \\\end{matrix}\right\}&m-i\\ \end{matrix}


A(x)=\sum_{i=0}^m a_{i}x^i\qquad \hbox{and} \qquad B(x)=\sum_{i=0}^n b_{i}x^i


\psi(t)= \begin{cases} 1 &\text{ if } t\geq 1,\\ 0&\text{ if }t\leq 0. \end{cases}


\mathcal{M}:=\{u\in X: \underline{u}\leq u\leq \overline{u} \text{ in }\Omega\}


\tilde C_2:=B_{d+1}\otimes \Lambda (V_1\oplus \overline{V}_1)\otimes \Lambda^+(V_2\oplus \overline{V}_2)


\chi \in B_{d+1}\otimes \Lambda V_1\otimes \Lambda^+ V_2


F_kC_1:=B_{d+1}\otimes \Lambda \overline{V}_1\otimes \Lambda^k V_1


{\underset {j \epsilon S} \oplus}{\bm e}(\tfrac{1}{n},j)({\bf A}{\bf w}(f,g))


\overset {n} {\underset {j=1} \oplus} {\bf A}{\bf w}(f,g) {\bf e}(\tfrac{1}{n},\xi) ={\bf A} {\bf w}(f,g)


S \subseteq \{1,2,\dots, n\} \times \{1,2,\dots, f+g\}


\Theta(x) := \frac{\partial Z}{\partial x}(x)\begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} H(x) P^{-1} \\ G(x)Y(x) \end{bmatrix}


\frac{\partial V}{\partial x}(x)(AF(x) + BG(x)K(x)) = 2Z^{\top}(x) P \Theta(x) PZ(x)


\tilde{L}=\{x\in \mathbb{R}^n: \Phi(x)\leq \Phi(x^k),~k=0,1,2,\ldots\}


q: [0,1] \to \operatorname {Ham} (M,\omega)


\phi _{\mathcal{A}}: [0,1] \to Ham (M,\omega)


(x_0, \alpha_0, \mu) \mapsto \partial_{\alpha_0}(b_0, b, f_0)


(x, \alpha, \mu) \mapsto \partial_\alpha(b, f)


(x, \alpha, \mu) \mapsto \partial_x(b, \sigma, f)


(x_0, \alpha_0, \mu) \mapsto \partial_{x_0} (b_0, b, f_0)


\Phi: \left( \bigoplus_{i=0}^{n-1} C^*(\Omega_1^{w_i}) \right)\rtimes_{\rho} \Z^{k_2} \to C^*(\Omega)


\lambda \in \Omega_1^{w_m} (\cong \Lambda )


\left( \bigoplus_i C^*(\Omega_1^{w_i})\right) \rtimes_{ \rho} \Z^{k_2}


\log |F (1/2 + it)| \approx \Re \sum_{p \leq \sqrt{x}} \frac{f(p)}{p^{1/2+it}}


\frac{{(\log \log x )}^{1/2}}{\log x} \int_{\R} \biggl| \frac{F_{(x)} (1/2 + it)}{1/2 + it} \biggr|^2 \, dt \xrightarrow{d} V_{\mathrm{crit}} \quad \text{as } x \rightarrow \infty


F_{(x)} (s) \coloneqq \prod_{p \leq x} {\bigl( 1 - \frac{f(p)}{p^{s}} \bigr)}^{-1}


\alpha \in \mathbb{Z}^{d}, r>0, Q(\alpha,r)=\alpha+(-r/2,r/2)^{d}


\mathcal{A}_{\varepsilon}(\Omega;\mathbb{R}^{m})=\{u:\mathbb{R}^{d} \rightarrow \mathbb{R}^{m}: u\ \text{constant on}\ \alpha +[0,\varepsilon)^{d}\ \text{for any}\ \alpha \in \Omega_{\varepsilon}\}


\alpha(t)\nabla u_1(t)\cdot\nu_i(t)+s_0u_1(t)=\alpha(t)\nabla u_2(t)\cdot\nu_i(t)+s_0u_2(t)\quad\text{on }\Gamma(t)\text{ for }i=1,2


\{\pi_{3i+1}\mid 0\le i \le \lfloor p/3 \rfloor-1 \}


\rho(G)\ge \lfloor p/3 \rfloor \geq \lfloor \gamma(G)/3 \rfloor


(v, w) \in V \times L^\infty(\Gamma_\mathrm{T})


\epsilon : H^1(\Omega,\mathbb{R}^3) \to L^2(\Omega,\mathbb{R}^{3\times 3}_{\mathrm{sym}})


\tilde{n} \colon \Gamma_\mathrm{T} \to \mathbb{R}^3


K = \big\{ u \in V \mid u \cdot \tilde{n} \leq \phi_1 \text{ on } \Gamma_\mathrm{T} \big\} \,


(u^{k}, \psi^{k}) \in V \times L^\infty(\Gamma_\mathrm{T})


[\text{Sv} \cdot \text{m}^2/(\text{Bq} \cdot t)]


L_{a_1,\ldots, a_n} \oplus L_{a_1,\ldots, a_n}


{\bar {\mathcal{A}}}_u \in \mathbb{R}^{nm \times nm}


{\bar {\mathcal{B}}}_y \in \mathbb{R}^{np \times p}


{\bar {\mathcal{A}}}_y \in \mathbb{R}^{np \times np}


{\bar {\mathcal{B}}}_u \in \mathbb{R}^{nm \times m}


\lim_{k \rightarrow \infty} \hat x(k) = x(k)


({\bar {\mathcal{A}}}_u , ~{\bar {\mathcal{B}}}_u)


({\bar {\mathcal{A}}}_y , ~{\bar {\mathcal{B}}}_y)


g = U^{\frac{4}{n-4}} \overset{\circ}{g}


u:\R^n \backslash \{ 0 \} \rightarrow (0,\infty)


f^2=(h + \overline{g})^2 = h^2 +\overline{g}^2+ 2h\overline{g}


\frac{\partial h}{\partial z}= 0, ~~ ~~ \frac{\partial \overline{g}}{\partial\overline{z} } =0


\frac{\partial^2 f^2}{\partial z\overline{z}} = 0


\frac{\partial^2 f^2}{\partial z\overline{z}} = 2\frac{\partial h}{\partial z}\frac{\partial \overline{g}}{\partial\overline{z} } =0


\sigma_1(\overrightarrow f_{j})= \overleftarrow f_{j}; 1\leq i \leq 3 \text{ and } 1 \leq j \leq 7


H(Z) = \mathbb{E}[I(Z)] = \sum_{z} P(z) I(z)= - \sum_{z} P(z) \log P(z)


\log P(z) = \log |P(z)| + i \theta(P(z))


U_p : (s,\infty) \rightarrow L^2_\omega(\mathbb{R}^n)


\nabla_x U_p(t) \in L^2_\omega(\mathbb{R}^n)


E_{\alpha}^{\sigma}(\mathbf{x}; q, t) = E_{\alpha}^{\sigma s_i}(\mathbf{x}; q, t)


\forall x\in (u^{-1}(0))^{\alpha_0}, \qquad |\nabla u(x)|\ge \frac{k}{2}


D(S(\lambda)) := \set{u\in W^{1,\infty}[-\tau,\tau]\mid u(-\tau)\in N(P_\lambda^-(-\tau)), u(\tau)\in R(P_\lambda^+(\tau))}


% H^1_{\Theta}=K^1_{\Theta}\oplus \Theta H^1


H^1_\Theta/(K^1_{\Theta}\oplus \Theta H^1 )


K^1_{\Theta}\oplus \Theta H^1 \subsetneq H^1_{\Theta}


\varphi_{\lambda}(x) = \varphi(V(I-\overline{\lambda}V)^{-1}x)=\varphi(Rx) = 0


\langle x'', x' \rangle \ge \gamma > \langle x'', x'_0 \rangle


\langle x, x' \rangle \ge \gamma > \langle x, x'_0 \rangle


g_{ij}=\langle \partial x_i, \partial x_j\rangle


v_p^T g \mathcal{A}_{X,p}v_p \leq -\alpha_p v_p^T \mathcal{A}_{X,p}^T g \mathcal{A}_{X,p} v_p,\quad\forall v_p\in\mathbb{R}^d


\langle \nabla_{r_p} X, r_p\rangle + \beta\langle \nabla_{r_p} X, n_p\rangle \leq -\alpha\|\nabla_{r_p}X\|^2


-\alpha_p = \max_{1\leq i \leq d} \lambda_i\left(\frac{M+M^T}{2}\right)


\|[b,T_\Omega]\vec f-[b,T_{\Omega,\delta}]\vec f\|_{L^p(W)}\lesssim_{b,\Omega}\delta\, \|M_W(W^\frac{1}{p}\vec f)\|_{L^p(\R^d)}\lesssim_{W} \delta\,\|\vec f\|_{L^p(W)}


I_t^{\mathbf{Y}} = \int_0^t (h,(D_x h f + D_y h)) (s, X^{\mathbf{Y}}_s, Y_s) d \mathbf{Y}_s - \frac{1}{2} \int_0^t | h (s, X^{\mathbf{Y}}_s, Y_s) |^2 ds


Y_t = \int_0^t h (X_s, Y_s) ds + B^{\perp}_t


T_0\coloneqq \{t\in T\mid \forall\chi\in X^*(T),\, \omega(\chi(t))=0\}


\zeta_\alpha\left(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\right)


T_r\coloneqq \{t\in T_0\mid \forall\chi\in X^*(T),\omega(\chi(t)-1)\geq r\}


(st)^{+} = s^{+} \quad \text{and} \quad (st)^{\ast} = t^{\ast}


S_k=0, \ \text{if}\ k \in \partial\mathcal{L}


\mathbf{r}_i = \begin{bmatrix} b_{i,1}, & b_{i,2}, & \cdots, & b_{i,n} \end{bmatrix}


b_{i,j} - \infty = \infty - b_{i,j} = \infty - \infty := -\infty


z_{k,c}^{'}=\mathbf h_{k,c} \sum\limits_{l \ne k}^K \mathbf p_{l}^c x_{l,c} + z_{k,c}


\mathbf{A} = \rm{Bdiag}\{\mathbf{A}_1, \cdots, \mathbf{A}_N\}


\mathbf p_{k}^c \in \mathbb{C}^{M \times 1}


{\mathcal{R}_{k,c}} = \mathrm{log}(1+{\Gamma}_{k,c}^{-1}\mathbf{h}_{k,c}\mathbf p_k^c(\mathbf p_k^c)^H \mathbf{h}_{k,c}^H)


\mathbf H_k={\rm Bdiag}\{\mathbf h_{k,1},\cdots,\mathbf h_{k,N_v}\}\in \mathbb{C}^{N_v \times MN_v}


\mathbf x_{k,c}= \sum\limits_{k=1}^K \mathbf p_{k}^c x_{k,c}


d_G(v,u)< \lfloor\frac{3}{2}K\rfloor+K+ \lfloor\frac{3}{2}K\rfloor\le 4K


\widetilde{K} = \widetilde{P_1} \times \widetilde{P_3}


\widetilde{K} \cap \widetilde{P_1} \neq 1


\widetilde{K} \cap (\widetilde{P_1} \times \widetilde{P_2})


\widetilde{K} \cap (\widetilde{P_1} \times \widetilde{P_2}) \lhd \widetilde{P_1} \times \widetilde{P_2}


T_k : \mathcal{S} \rightarrow \mathcal{S}


\begin{aligned} y \cdot \frac{1}{z^d} &=\int \frac{\alpha}{z} \cdot \frac{1}{z^d} d z+\beta=-\frac{\alpha}{d z^d } +\beta, \end{aligned}


R(z)=\frac{z^d}{ \beta z^d -\frac{\alpha}{d}}


\left\{\mathbf{g}_1, \mathbf{g}_2, \ldots, \mathbf{g}_\rho\right\} \subseteq V


\Lambda = (\lambda_1, \lambda_2, \ldots, \lambda_\ell)


\left(\lambda_1^{-1}, \lambda_2^{-1}, \ldots, \lambda_\ell^{-1}\right)


V=\bigoplus_{i=1}^\ell\frac{\mathbb{F}_q[x]}{\langle x^{m_i}-\lambda_i\rangle}=\frac{\mathbb{F}_q[x]}{\langle x^{m_1}-\lambda_1\rangle}\oplus \frac{\mathbb{F}_q[x]}{\langle x^{m_2}-\lambda_2\rangle}\oplus \cdots \oplus \frac{\mathbb{F}_q[x]}{\langle x^{m_\ell}-\lambda_\ell\rangle}


d_{i+1,N}\leq 5 \cdot 2^{N-i} \cdot 2^i \cdot d_{i,N}=5 \cdot 2^N\cdot d_{i,N}


e \, c \, d \, \bar c \, \bar d \, \bar e \, a \, b \, \bar a \, \bar b \, e


D(\text{std}(\text{conv}(w))) \subset [k] \times [n]


v^Td<b~\forall d\in Conv\bigg\{\displaystyle \bigcup_{j\in \Lambda}\partial H_j(x^*)\bigg\}


Conv\bigg\{\displaystyle \bigcup_{j\in \Lambda}\partial H_j(x^*)\bigg\}


0\not\in Conv\bigg\{\displaystyle \bigcup_{j\in \Lambda}\partial H_j(x^*)\bigg\}


\bm{c} \in H^1_{\text{div}}(\Omega) = \{\bm{c} \in [L^2(\Omega)]^2 \ | \ \nabla \cdot \bm{c} \in L^2(\Omega)\}


H^r_D(\Omega) = \{u \in H^r(\Omega) \ : \ u\big|_{\Gamma_D} = 0\}


BS_n\backslash (BS_{n-1}^1\oplus BS_{n-1}^2\oplus BS_{n-1}^3)


S_{i j}^{\prime}=S_{i j} \cap I_{j}=\left\{L_{j}\right\}


\int_{\Gamma_\sigma}g(d(x))\ dx %=\int_{\Gamma_\sigma}g(d(x))\ |\nabla d(x)| dx =\int_0^\sigma g(t)\mathcal H^{N-1}(\Gamma_\sigma\cap\{d=t\})\ dt


c_1H^{N-1}(\partial \Omega)\le H^{N-1}(\Gamma_\sigma\cap\{d=t\})\le c_1H^{N-1}(\partial \Omega)


\sup_{B\subset\Omega}\left(\frac1{|B|}{\int_Bw\ dx }\right)\left(\frac1{|B|}{\int_Bw^{-1}}dx \right)<+\infty


\eta(f, g)\colon \begin{array}{ccc} \Omega^1_0(M) & \to & \Omega^1_0(M)\\ \zeta &\mapsto & \zeta^*[f,\tau^*\zeta^*g] \end{array}


\widetilde{G}(tX,t^\beta Y) = t^{d-e}\widetilde{G}(X,Y)


F(tX,t^\beta Y) = t^d F(X,Y)= t^d X^e \widetilde{F}(X,Y)


\widetilde{F}(tX,t^\beta Y) = t^{d-e}\widetilde{F}(X,Y)


F(tX,t^\beta Y) = (tX)^e \widetilde{F}(tX, t^\beta Y) = t^e X^e \widetilde{F}(tX,t^\beta Y)


\|\lambda a(\omega)+\nu b(\omega)\|=\|\lambda a(\omega)\|+\|\nu b(\omega)\|


%\label{eq: ||int_(E_i) z dmu|| > (1-eps) int_(E_i) ||z|| dmu} \begin{aligned} \biggl\|\int_{E_i} z\,d\mu\biggr\| &\geq\re\zs_i\biggl(\int_{E_i} z\,d\mu\biggr) =\int_{E_i}\bigl(\re\zs_i(z_i)+\re\zs_i(z-z_i)\bigr)\,d\mu\\ &\geq\int_{E_i}(\|z_i\|-\|z-z_i\|)\,d\mu \geq\int_{E_i}(\|z\|-2\|z-z_i\|)\,d\mu\\ &>\int_{E_i}(\|z\|-\eps\sigma)\,d\mu \geq\int_{E_i}(\|z\|-\eps\|z\|)\,d\mu =(1-\eps)\int_{E_i}\|z\|\,d\mu. \end{aligned}


\mathrm{\delta} = \frac{\rho_1 e^{j\omega_1}}{\rho_2 e^{j\omega_2}}


\mathrm{\textbf{H}^{'}_{2,D}} = \frac{\mathrm{\textbf{H}_{2,D}}}{\rho_2 e^{j\omega_2}}


\mathrm{\textbf{H}} = [\mathrm{\textbf{H}^{'}_{1,U}},\mathrm{\textbf{H}^{'}_{2,D}} ]


w_\perp^T A(\sigma)w_\perp -tw_\perp M w_\perp\geq \alpha w_\perp^T \overline{A}w_\perp -tw_\perp M w_\perp\geq (\alpha \rho \Lambda-t)\|w_\perp\|^2_M>0


-W_\perp (W_\perp^T (A(\sigma)-\lambda(\sigma)M)W_\perp)^{-1}W_\perp^T \delta A(\sigma)\overline{x}(\sigma)=W_\perp \gamma(\sigma)


Z(\sigma, t ) : S \times (0,\Lambda) \mapsto \mathbb{R}^{n\times n}


\theta\coloneqq (\hat d_D, \hat d_T, \hat r_R, \hat f_0)


\Delta: L(V,\mathbb{R}) \to L(V,\mathbb{R})


A \coloneqq (\omega(x,y))_{x,y\in V}\in \mathbb{R}^{|V| \times |V|}


\widehat{f} (\chi^{-1}) = \overline{\widehat{f}(\chi)}


\omega: V \times V \rightarrow \mathbb{R}_{\geq 0}


0 = \lambda_1< \lambda_2 \le \cdots \le \lambda_n


d_x \coloneqq \sum_{y \in V,\; y \sim x} \omega(x,y)


\partial\Omega_d = \partial\Omega \setminus \partial\Omega_n


\gamma_1, \gamma_2\colon \mathbb{R}\to \mathbb{C}


|W| = \left| \bigcup_{i = 1}^k W_i \right| < \alpha N / 64


|W_i| = \sqrt{\frac{2 K N \log N}{d}} =: \ell


\theta_{A,B,x}(s\otimes t)=\llbracket s\otimes t\rrbracket


k \otimes_{\mathbb{Z}}T_i =0 =\mathrm{Tor}_1^{\mathbb{Z}}(k,T_{j})=0\;, \text{ for $0<i<e$ and $0<j<e-1$.}


(a \star b) \star c = a \star (b \star c)


A = A \star A = A \star A \star A' = A \star A'


G = \bigsqcup_{A, B \in \mathcal{C}} \mathrm{Hom} (A, B)


g^{- 1} \star g = g^{- 1} \star g \star A' = A'


U_\omega^{(L)} = U_\omega^{\Lambda_L} \oplus U_\omega^{\Lambda_L^C}


\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/8 \mathbb{Z}


G_F(K)=\begin{bmatrix} l & m \\ m & n \end{bmatrix}


\frac{1}{2}|\!\operatorname{Tr} gh| = \left|\sinh\left(\frac{t_g}{2}\right)\sinh\left(\frac{t_h}{2}\right) + \cosh\left(\frac{t_g}{2}\right)\cosh\left(\frac{t_h}{2}\right)\cos\theta_P\right|


\operatorname{PSL}^{\pm}_2(\mathbb{R}) = \left\{ \pm \begin{pmatrix} a & b \\ c & d \end{pmatrix} \middle| a, b, c, d \in \mathbb{R}, ad - bc = \pm 1 \right\}


\{v\in V(G)\setminus X \, \colon v \text{ is adjacent to a vertex in } X\}


\{v \in V(G) \setminus D \, \colon N(v) \cap D = \{u\}\}


\mathcal L= \sum_{i,j}\left(\tilde p_{ij}\log\frac{\tilde p_{ij}}{p_{ij}}+\lambda_i(\tilde p_{ij}-\tilde \alpha_i)+\gamma_j(\tilde p_{ij}-\tilde \beta_j)\right)


(\lambda,x) \in [\underline{H},\overline{H}] \times \mathbb{R}


\frac{\partial H^h}{\partial l_i} = - K_i, \quad e_i\in E^h


\xi_k = \|\boldsymbol{x}_k-\boldsymbol{x}_*\|,\quad f(k) = -\ln \xi_k


\lim_{k\to\infty}\frac{\xi_{k+1}}{\xi_k^q}=Q_q, \quad q>1,\quad 0<Q_q<\infty


\lim_{k\to\infty} d(k) = -\ln Q_q \coloneqq d


S \coloneq \left\{ \{b_i\}_{i=1}^\infty \left\vert 0\leq b_i\leq a_i, \sum_{i=1}^\infty b_i = 1\right.\right\}


f(x) = \begin{dcases} 1/q & \text{ if } x=p/q\text{ with } (p,q)=1;\\ 0 & \text{ if } x \text{ is irrational;} \end{dcases}


Q=\begin{bmatrix} 10& 0\\ 0& 50\\ \end{bmatrix}


\Theta: {\Sigma_\vartheta} \times \mathbb{T} \to {\Sigma_\vartheta} \times \mathbb{T}


(x,y) \mapsto (T_\omega (x) , T_\omega(y))


\Theta^{(2)} : {\Sigma_\vartheta} \times \mathbb{T}^2 \to {\Sigma_\vartheta} \times \mathbb{T}^2


T^n_\omega (x) \coloneqq T_{\omega_{n-1}} \circ \cdots \circ T_{\omega_0} (x)


\int_{\Sigma_\vartheta} \mu^{(2)} \left( \left( T^{(2)}_\omega\right)^{-1} (A) \right) \, d\mathbb{P} (\omega) = \mu^{(2)}(A)


\sigma \omega \coloneqq (\omega_{i+1})_{i\in\mathbb{N}}


0\neq(a_1 \wedge a_2) \wedge n_i = (a_1 \wedge a_2) \wedge (n_i^1 \wedge n_i^2) =(a_1 \wedge n_i^1) \wedge (a_2 \wedge n_i^2)


a_1 \wedge \left( \bigwedge_{i=1}^2 n_i^1 \right) \neq 0\quad \text{and}\quad a_2 \wedge \left( \bigwedge_{i=1}^2 n_i^2 \right) \neq 0


(a_1 \wedge a_2) \wedge \left( \bigwedge_{i=1}^2 n_i \right) = (a_1 \wedge a_2) \wedge \left( \bigwedge_{i=1}^2 (n_i^1 \wedge n_i^2) \right) \neq 0


% \left\{ \left( 0_{X^{\prime }},0\right) \right\}


\left\{ x\in X:f\left( x\right) \geq 0\right\}


(\theta, \phi) \in \left[0, 2 \pi\right) \times [0, \pi]


E\in \big| \omega_X^{\otimes k} \otimes \alpha \big|


E'=R_k(E) \in \big| \omega_Y^{\otimes k} \otimes \beta \big|


\mathrm{ D}^b(\mathrm{ Unip}(X)) \simeq \mathrm{ D}^b ( \mathrm{ Coh}_{\{ \hat{0} \}}(Y) )


\mathrm{ D}_{ \mathrm{ Supp}(E) }^b(\mathrm{ Coh}(X))\simeq \mathrm{ D}_{\mathrm{ Supp}(E')}^b(\mathrm{ Coh}(Y))


H^0(X,\omega_X^{\otimes k_0}\otimes \alpha) \neq 0


R \colon \mathrm{ Aut}^0(X) \times \mathrm{ Pic}^0(X) \to \mathrm{ Aut}^0(Y) \times \mathrm{ Pic}^0(Y)


\nabla D^m\psi(x_1,\dots,x_m)=\left\{\nabla \psi(x_i) - \nabla \psi\left(-\sum_{i=1}^m x_i\right) \right\}_{i=1}^m


V_\epsilon^\star = \frac{|x|^2}{2} - \epsilon \psi(x) + \frac{\epsilon^2}{2}|\nabla \psi|^2 + o(\epsilon^3)


|\nabla D^m\psi(x_1,\dots,x_m)|^2=\sum_{i=1}^m\left|\nabla \psi(x_i) - \nabla \psi\left(-\sum_{i=1}^m x_i\right)\right|^2


x_0\in B_{\eta}\cap K\implies \lim_{t\to\infty}\|x(t;t_0,x_0)\|=0


x_0\in B_{\delta}\cap K\implies \|x(t;t_0,x_0)\|<\varepsilon


\lambda' = \lambda^{( k-1)} + \lambda^{( k)} + \cdots


\mathrm{exp} : ((\mathfrak{g}_A)_0, \mathrm{BCH}, 0) \cong (\Gamma_A, \circledcirc, 1) : \mathrm{log}\


1 \le \ell \le \lfloor \frac{p-k-1}{2k-1} \rfloor


|A| = 6 \lfloor \frac{p}{11} \rfloor - 3


|B| = k \lfloor \frac{p-k-1}{2k-1} \rfloor + 2


\alpha=\frac{A}{h_{0}}\ll 1, \qquad \beta=\frac{h_{0}}{l^{2}}\ll 1, \qquad S=\frac{\alpha}{\beta}=\frac{Al^{2}}{h_{0}^{3}}\approx 1


u, v \in C^{1+\beta, \frac{1+\beta}{2}}(\Omega_{T_m-\varepsilon})


\|u\|_{C^{2+\beta,1+\frac{\beta}{2} }(\Omega_{T_m - \varepsilon} \setminus \Omega_{T_0})} \leq Q^*


\underline{u}(0, x) \leq u_0(x) \quad \text{for } x \in [-h_0, h_0]


\textsf{Sfas}(T_{\sigma}) = \textsf{Sfas}(T_{\sigma} [A_I(T_{\sigma}, P_{\sigma})])+\textsf{Sfas}(T_{\sigma} [A_B(T_{\sigma}, P_{\sigma})])


\textsf{Sfas}(T_{\sigma}) \leq \textsf{Sfas}\big(T_{\sigma}[A_I(T_{\sigma}, P_{\sigma})]\big) + \textsf{Sfas}\big(T_{\sigma}[A_B(T_{\sigma}, P_{\sigma})]\big)


\textsf{Sfas}(T_{\sigma}) \geq \textsf{Sfas}(T_{\sigma}[A_1]) + \textsf{Sfas}(T_{\sigma}[A_2])


\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}


B_\infty=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}


\tilde A(t)=t^\alpha B+o(t^\alpha) \in V(\mathbb{K})\cap PSL_2(\mathbb{K})


\mathbb{K}P^1\ni[z:w]\longmapsto Z(t)=\begin{bmatrix} tw & z \\ 0 & t^{-1}w \end{bmatrix}\in\overline{\operatorname{PSL_2}\mathbb{K}}=\mathbb{K}P^3


\begin{bmatrix} 0 & ct^\gamma \\ 0 & 0 \end{bmatrix}


\begin{bmatrix} t & ct\\ 0 & 0 \end{bmatrix}


\langle f_\alpha : \alpha < \omega_{n+1} \rangle


\gamma = \sup\{\beta_\eta : \eta < \lambda^{+k}\}


\langle f_{\alpha} : \alpha < \omega_{1} \rangle


\{t\in[0,\infty):f\textrm{ is discontinuous at }t\}=\{f\neq f_{-}\}\cup\{f\neq f_{+}\}


\mathrm{co}\left\{\left\vert{\int_{0}^{t}\xi dX^{n}}\right\vert:n\in\mathbb{N},\xi\textrm{ is predictable and }\vert{\xi}\vert\leq1\right\}


\left\{ \begin{aligned} \frac{d}{dt}u + vAu &= B(u_{2})-B(u_{1})\\ u(0) &= 0 \end{aligned} \right. ,\quad u \in L^{\infty}(0,T;V) \cap L^{2}(0,T,D(A))


\frac{d}{dt}\Vert u \Vert_{H}^{2} \leq \frac{c^{2}}{v}\Vert u \Vert_{H}^{2}\Vert u_{2} \Vert_{V}^{2}


\left\{ \begin{aligned} (B(u_{1},u),u) &= 0\\ \vert (B(u,u_{2}),u) \vert &= \vert \int_{\Omega} u \cdot \nabla u_{2} \cdot u dxdy\vert \leq \Vert u \Vert_{L^{\infty}}\Vert \nabla u_{2} \Vert_{V} \Vert u \Vert_{L^{2}} \end{aligned} \right.


\left\langle \{t_{ij}^k\}_{i,j \in I}^{1\leq k\leq d}\right \rangle


\ket{0}=\begin{bmatrix} 1\\0 \end{bmatrix},\quad \ket{1}=\begin{bmatrix} 0\\1 \end{bmatrix}


\omega(\cdot, \cdot): \mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}


f(\cdot ) : \mathbb{R}^n\rightarrow \mathbb{R}^n


h(\cdot ) : \mathbb{R}^n\rightarrow \mathbb{R}


S(\cdot) : \mathbb{R}^n\rightarrow \mathbb{R}


g(\cdot ) : \mathbb{R}^n\rightarrow \mathbb{R}^n


\mathbf{r}_t = \sigma(\mathbf{W}_r \mathbf{x}_t + \mathbf{U}_r \mathbf{h}_{t-1} + \mathbf{b}_r)


\mathbf{h}_t = (1 - \mathbf{z}_t) \odot \mathbf{h}_{t-1} + \mathbf{z}_t \odot \tilde{\mathbf{h}}_t


\tilde{\mathbf{h}}_t = \tanh(\mathbf{W}_h \mathbf{x}_t + \mathbf{U}_h (\mathbf{r}_t \odot \mathbf{h}_{t-1}) + \mathbf{b}_h)


\mathbf{h}_t = f(\mathbf{h}_{t-1}, \mathbf{x}_t; \theta)


\mathbf{z}_t = \sigma(\mathbf{W}_z \mathbf{x}_t + \mathbf{U}_z \mathbf{h}_{t-1} + \mathbf{b}_z)


y_i = x_i^{\top} \beta_0 + \varepsilon_i, \quad i=1,\dots,n


e\in \operatorname{int}\left(\mathcal{B}_{G} \right )


\mathcal{F}: \mathbb{R}^d \to \mathbb{R}^{n}


\pi(x \mid y)=\frac{\pi(y| x)\pi_0(x)}{Z(y)}\propto \exp(-\Phi(x))\pi_0(x)


\Vert J_i(\mathbf{u})-J_i(\mathbf{v})\Vert \leq l_J\|\mathbf{u}-\mathbf{v}\|


\mathbf{x}^*=\mathrm{col}(x_1^*,x_2^*,...,x_N^*) \in \Omega \cap \Upsilon


J_i(x_i^*,h(\mathbf{x}^*)) \leq J_i(x_i,\frac{1}{N}x_i+\sum_{j \neq i} \frac{1}{N}x_j^*), \nonumber


\mathcal{T}_{\alpha}: \mathbb{R}^{M_{1}\times M_{2}\times\dots\times M_{N}}\rightarrow \mathbb{R}^{M_{1}\times M_{2}\times\dots\times M_{N}}


\sum_{n=1}^N \sum_{m=1}^N \int_0^T a_n a_m w(nf,t) w(mf,t)\,dt


\left|\int_0^T w_c(f,N,A,t)^2\, dt - \frac{NT}{2}\right| \le \frac{N^2}{2\pi f}


\mathcal{I} = (I, \mathcal{E}^I, \mathcal{P}(I))


I_C = \{i \in I \mid A_i \cap \operatorname{dom}_1(C) \neq \emptyset\} \subseteq I


I = E[I_C] = \bigcup_{i \in I_C} E[\{i\}]


|I| = |I| \times \aleph_0 = |J| \times \aleph_0 = |J|


\int_{I}\left|f_n\left(t_0\right)(x)-f\left(t_0\right)(x)\right| d x \geq \int_{\tilde{I}}\left|f_n\left(t_0\right)(x)-f\left(t_0\right)(x)\right| d x>\epsilon_0 \cdot m(\tilde{I})


C\left([0, T] ; L^2(I ; \mathbb{R} \otimes \mathbb{R})\right)


\mathbf{H}_{r}^{\text{NLoS}} \in \mathbb{C}^{M_r \times N}


e^{j 2 \pi d_{\text{R}} \sin(\gamma) \left[ \lfloor \frac{n-1}{N_x} \rfloor \sin(\zeta) + ((n-1)-\lfloor \frac{n-1}{N_x} \rfloor N_x) \cos(\zeta) \right]/\lambda }, n = 1, 2, \cdots, N


\mathbf{n}_{\text{BS},i} \sim \mathcal{CN} \left( \mathbf{0}_{M_t}, \sigma_{\text{BS}}^2 \mathbf{I}_{M_t} \right)


\boldsymbol{\varphi} = [\varphi_1, \varphi_2, \cdots, \varphi_N]^T


\mathbf{H}_t \in \mathbb{C}^{N \times M_t}, \mathbf{H}_r \in \mathbb{C}^{M_r \times N}


\mathbf{H}_d \in \mathbb{C}^{M_r \times M_t}


K_{ab}{}^I = - u^\mu_a u^\nu_b \nabla_{(\mu} n^I_{\nu)} =n^I_\mu D_a u^\mu_b + \frac{1}{2} \hat \upsilon^I \tau_{ab}


\mathcal{R}^a{}_{bcd}= \partial_c \gamma^a_{d b} - \partial_d \gamma^a_{c b} + \gamma^a_{cf} \gamma^f_{d b}- \gamma^a_{df} \gamma^f_{c b}


\mathcal{D}_a T^I = D_a T^I - \omega_a{}^I{}_J T^J


\omega_a{}^I{}_J = n^\mu_J D_a n^I_\mu \,


2 \gamma^c_{[ab]} = -\hat \upsilon^c F_{ab}


\delta K_{ab}{}^I = \frac{1}{2} F_{I a} \partial_b \sigma + \frac{1}{2} F_{Ib} \partial_a \sigma


Z=\left( \begin{matrix} I_{d\times d}&0\\ (z_0,z_1,...,z_{d-1})&1 \end{matrix} \right)


\begin{aligned} \mathcal{L}_{\text{SIMax}} = \mathbb{E}_{z \sim {p_{\theta}(z|x)}} \{ & \mathbb{E}_{t \sim T} \left[-\log q_{\phi_t}(y_{t}|z)\right] \\ & - \alpha \log \frac{\exp(\text{sim}(z, z^{+}) / \tau)}{\sum_{z' \in \mathcal{B^{+}}} \exp(\text{sim}(z, z') / \tau)} \}, \end{aligned}


Y_t \rightarrow X \rightarrow Z \rightarrow Z_t \rightarrow \hat{Y}_t


H(x) = \sum_{i \neq j} Q_{ij} x_i x_j + \sum_i q_i x_i


\begin{aligned} H(\sigma) &= \sum_{i \neq j} Q_{ij} \cdot \frac{(\sigma_i + 1)(\sigma_j + 1)}{4} + \sum_i q_i \cdot \frac{\sigma_i + 1}{2} \\ &= \sum_{i \neq j} \frac{Q_{ij}}{4} \sigma_i \sigma_j + \sum_i \left( \frac{q_i}{2} + \frac{\sum_{i \neq j} (Q_{ij} + Q_{ji})}{4} \right) \sigma_i \\ &\quad + \left( \frac{\sum_i q_i}{2} + \frac{\sum_{i \neq j} Q_{ij}}{4} \right) \end{aligned}


\begin{aligned} Q_{ij} = & \, P_1 \sum_{i \neq j} x_{S,i} x_{S,j} + P_2 \sum_{i \neq j} x_{i,D} x_{j,D} \\ & + P_3 \sum_{i \not\in \{S,D\}} \left( \sum_{(i,j) \in E} \sum_{\substack{(i,k) \in E \\ k \neq j}} x_{i,j} x_{i,k} \right. \\ & \quad + \left. \sum_{(j,i) \in E} \sum_{\substack{(k,i) \in E \\ k \neq j}} x_{j,i} x_{k,i} - 2 \sum_{(i,j) \in E} \sum_{(k,i) \in E} x_{i,j} x_{k,i} \right) \end{aligned}


H(\sigma) = -\sum_{i,j} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i


\begin{aligned} q_i = & -P_1 \sum_i x_{S,i} - P_2 \sum_i x_{i,D} \\ & + P_3 \sum_{i \not\in \{S,D\}} \left( \sum_{(i,j) \in E} x_{i,j} + \sum_{(k,i) \in E} x_{k,i} \right) + f(x) \end{aligned}


A_n: f(x,y,z)=xy-z^{n+1}, \ \ n\in \mathbb{N}


\bar{f}_s \in U_{ad} \subset L^2(\Omega)


\begin{cases} (-\Delta)u = f , & x\in \Omega, \\ u=0, & x\in \partial \Omega, \end{cases}


\prod_{j = 1}^{P_{\Delta_n}} F_{i, j} \leq c_{i, \Delta_n}^{P_{\Delta_n}}


\prod_{j = 1}^{P_{\Delta_n}} F_{i, j} \leq b_{i, \Delta_n}^{P_{\Delta_n}}


\log \mathcal L = N \left( \alpha \log \lambda - \log \Gamma(\alpha) \right) + (\alpha-1) \sum_i \log \log \frac{\mathcal P_i}{\mathcal P_0} - \lambda \sum_i \log \frac{\mathcal P_i}{\mathcal P_0}


\mathrm{grad(x)_{i} \coloneqq \frac{\partial f(x)_i}{\partial x}}


\mathrm{SmoothGrad}(x) = \frac{1}{n}\sum_{i = 1}^{n} \nabla f(x + a)


\mathrm{Input*Gradient} \coloneqq x\odot \frac{\partial f(x)_i}{\partial x}


\liminf_{l \to \infty} \frac{\log(b_1 b_2 \dots b_l)}{l \log N}


\underline{V}_{\mathcal{L}} \in \mathbb{F}_{2}^{\ell \times n}


\mathbb{P}(\underline{\hat{V}}_{\mathcal{L}}(\underline{Y}_{\mathcal{L}}) \neq \underline{V}_{\mathcal{L}}) \leq \epsilon_e


K \subset \{\mu \in P(X \times Y): \mu_X^* \in P(X) \text{ is the } X\text{-marginal of } \mu\}


\int_{\R^n} Ta(x)x^{\alpha}dx=0, \quad \forall f\in L^2_{N_p}(\R^n) \text{ and } |\alpha|\leq N_p


\phi \mapsto (m \mapsto \langle m,\phi\rangle_{\text{\AE}})


m = \sum_{k=1}^{\infty} (\delta_{a_k} - \delta_{a}) \in \text{\AE}(X)


\underline{\varepsilon} \cdot \underline{x} = \sum_{i=1}^{5} \varepsilon_i x_i \leq 1, \quad \underline{x}\in \mathbb{H}_5


g_i\ge\prod_{\substack{p~\text{inert in~}K\\(p+1)\mid M_{x_i}}}2>\exp\left((\log2)\exp\left(C \frac{\log x_i}{\log\log x_i}\right)\right)>i


L(g_i)\le M_{x_i}\le x_i^2=(\log i)^{(4/C)\log\log\log i}<(\log g_i)^{c_0\log\log\log g_i}


|\alpha(x)| \lesssim_N \langle x\rangle^{-N}


\beta(\varphi)=\sup_{\mu\in \mathcal{M}_T(X)}\int \varphi\ d\mu


\overline{B}^\nu_{h,\alpha} := \big\{ r \in [1,\infty) : m^\nu([r, r+h]) \geq \frac{\alpha h}{ r}\big\}


H^{1}_{\pm}(K_{\infty, w}, E[p^{\infty}]) \coloneq \varinjlim H^{1}_{\pm}(K_{n, p}, E[p^{\infty}])


B_{v_{\infty}} \coloneq E[p^{\infty}]^{G_{K_{\infty, v_{\infty}}}}


f: \mathbb{R}^n \rightarrow \mathbb{R}^m


\nabla_{xx} f \in \mathbb{R}^{n \times n}


G_{\min}(x)=\min \left\{G_i(x)\mid i\in I\right\}


d((x,y),K)<\min \left\{d((x,y), L_i)\mid i\in I\right\}=d((x,y),\cup_{i\in I\ } L_i)=d((x,y), L)


\gamma = (\gamma_1, \gamma_2, \gamma_3) \in X


\varnothing \rightarrow u_1 \rightarrow u_1u_2 \rightarrow ... \rightarrow u_1...u_n=u


F_{\mu\nu}=\sum_k p_{\bm \lambda}(k)\,\partial_\mu \log p_{\bm \lambda}(k)\, \partial_\nu \log p_{\bm \lambda}(k) = \sum_k \frac{\partial_\mu p_{\bm \lambda}(k)\, \partial_\nu p_{\bm \lambda}(k)}{p_{\bm \lambda}(k)} \,


\bm E_\nu(\hat {\bm \lambda})=\hat{\bm \lambda}\, \qquad \partial_\mu E_k(\hat \lambda_\nu)=\delta_\mu^\nu


\mathbf V(\hat{\bm \lambda})\geq \frac{1}{M \mathbf F}\,


% \mathbf{a}(\theta)=\dfrac{1}{\sqrt M} \Big[1, \ e^{j % 2\pi\bar{d} \sin\theta}, \ \cdots, \ e^{j % 2\pi (M-1)\bar{d}\sin\theta}\Big]^\mathrm{T}\!\!,\raisetag{15pt} %


\mathbf{a}(\theta)=\frac{1}{\sqrt M} \Big[1, \ e^{j 2\pi\bar{d} \sin\theta}, \ \cdots, \ e^{j 2\pi (M-1)\bar{d}\sin\theta}\Big]^\mathrm{T}


\tilde{\mathbb{H}}^k\mathbb{E}^0 = (\mathbb{E}^0\otimes 1)\tilde{\mathbb{H}}^k


X(\theta_{i}): \begin{bmatrix}x_{i} \\ p_{i} \end{bmatrix} \mapsto \begin{bmatrix} \cos(\theta_{i}) & \sin(\theta_{i}) \\ -\sin(\theta_{i}) & \cos(\theta_i) \end{bmatrix}\begin{bmatrix}x_{i} \\ p_{i} \end{bmatrix}


\lambda_{i}^{H} \lambda_{n+1-i}^{V} \equiv \text{const}


\mathcal{E}(\mathbf{z},\epsilon) = x^2 + \epsilon^2 y^2 + p_{x}^2 + p_{y}^2


h_{mk}=\frac{\sqrt{\eta} e^{-2\pi j \left(\frac{ 1}{\lambda}\left| {\boldsymbol \psi}_m - {\boldsymbol \psi}_k^{\rm Pin}\right| +\frac{1}{\lambda_g}\left| {\boldsymbol \psi}_0^{\rm Pin} - {\boldsymbol \psi}_k^{\rm Pin}\right| \right)}}{ \left| {\boldsymbol \psi} _m - {\boldsymbol \psi}_k^{\rm Pin}\right|}


\mathcal{E}_0 = \mathbb{C}\,I_{\mathcal{H}}, \quad\text{and for each integer } t \ge 1


\{\mathcal{E}_t\}_{t \ge 0} \;\subseteq\; \mathcal{B}(\mathcal{H})


a^s(x,\bm{y})= \exp\left(\sum_{j=1}^{s}\frac{1}{j^2}\sin(2j\pi x)y_j \right)


x \in \complement \overline{V_i} \cap U_i


L_i(u,t) = \begin{cases} p_i(H(u,2t)) & \text{if } 0 \leq t \leq \frac12 \\ G(h_i(u), 2t-1) & \text{if } \frac12 \leq t \leq 1. \end{cases}


I(C_n^{\boxtimes d}, x) \equiv 1 + F(x) \equiv (x+1)^m \pmod{p}


I(C_n^{\boxtimes d}, x) \equiv c \cdot (x+1)^m \pmod{p}


I(C_n^{\boxtimes d}, x) \equiv 1 \pmod{n}


I(C_n^{\boxtimes d}, x) \equiv (x+1)^m \pmod{p}


F(x) = \sum_{S \in \mathcal{F}_p} x^{|S|}


E_\mathbf{I}(u\circ\eta|_{B(p, r_0)})\leq E_\mathbf{I}(u|_{S^2\setminus B(\bar{p}, L\bar{r})}) \leq \frac{\varepsilon_\mathbf{I}}{5}


\eta(B(p, r_0)) \subset B(\eta(p), r\kappa^2r_0) \subset B(\eta(p), r)


\eta(B(p, r_0))\subset S^2\setminus B(p_-, Lr)


\eta(B(p_+, 2r_1)) = S^2\setminus B(p_-, h(Lr))\subset S^2\setminus B(p_-, Lr)


L\coloneqq 2e^{\delta^{-1}k_\mathbf{I}(e_\mathbf{I}(\varphi) + 10^{-1}\varepsilon)}


x = \sqrt{\alpha P}s_1 +\sqrt{\bar \alpha P}s_2


\hat{\mathbf{s}} = g_{\theta'}(\mathbf{u}) = \sigma(W'\mathbf{u} + \mathbf{b}')


\sigma(\mathbf{x}) = [\sigma(\mathbf{x}_1), \dots, \sigma(\mathbf{x}_d)]^T


\mathbf{u} = f_\theta(\mathbf{s}) = \sigma(W\mathbf{s} + \mathbf{b})


\min_{\theta,\phi} \mathbb{E}_{\mathbf{y}, \mathbf{x}} l(f_\theta(\mathbf{x}_s,z),\mathbf{y})


\operatorname{GoE}_m(t) = \frac{u_m(t)}{\Delta_m(t)}, \forall m


v_{\rm cpt}(x) = \begin{cases} v^+_{\rm cpt}(x) = (x - x_{\rm ref})^{\alpha_{\rm cpt}}, ~x \geq x_{\rm ref};\\ v^-_{\rm cpt}(x) = -\lambda_{\rm cpt} (x_{\rm ref} - x)^{\beta_{\rm cpt}}, ~x < x_{\rm ref}, \end{cases}


g_m(t; y_m(t)) = \operatorname{min}\left\{ 1, \frac{y_m^{\alpha_m-1}(t) (1 - y_m(t))^{\beta_m-1}}{\operatorname{B}(\alpha_m, \beta_m)} \right\}


h^{\mathrm{BdG}}_8(\mathbf{k})=\frac{1}{2}\left[\sin k_x\sigma^x+\sin k_y \sigma^z +\sin k_z\tau^z\sigma^y\right]


\hat S_0(\textbf{k}) = \tau^z \qquad \hat{Q}_0=\sum_\mathbf{k} d_{\mathbf{k}}^\dag \hat S_0(\mathbf{k})d_{\mathbf{k}} \\ \hat S_1(\textbf{k}) = \cos k_z \tau^z + \sin k_z \tau^x \qquad \hat{Q}_1=\sum_\mathbf{k} d_{\mathbf{k}}^\dag \hat S_1(\mathbf{k})d_{\mathbf{k}}


\hat S_{1,K}(\textbf{k}) = \cos K\ \tau^z + \sin K\ \tau^x


\mathbf{V}=[ \mathbf{v}_1, \dots, \mathbf{v}_K] \in \mathbb{C}^{N\times K}


\mathbf{H}=[\mathbf{h}_1, \dots, \mathbf{h}_K] \in \mathbb{C}^{N\times K}


\varphi = (\varphi_1,\varphi_2, \varphi_3) \colon \tilde{U}(B) \rightarrow [X]


f \cdot g=(f \ast (f' \ast -), (- \ast f) \ast f'), f \ast [f',-] + [f,-] \ast f')


[x,yz]=[x,y]z+y[x,z], \quad \forall x,y,z \in X


\cdots\vert \;f \ast [x,y]=[f \ast x,y]-[f,y]x,  [x,y]*f=[x \ast f,y]-x[f,y],  [f,xy]=[f,x]y+x[f,y] \rbrace


\cdot \colon X \times X\rightarrow X \quad \text{ and } \quad [-,-] \colon X \times X \rightarrow X


(b,x) \cdot (b',x') = (bb', xx'+ \varphi(b)x'+x\varphi(b'))


\mathcal A\in\mathbb R^{n_1}\otimes\dots\otimes\mathbb R^{n_k}


G(x) \geq 0, \; H(x) \geq 0, \; G_i(x)H_i(x) = 0, \quad \forall i = 1 \dots n_c


\hat{\lambda_l} = \sum^K_{k=1} \lambda_{l,k}, \quad \hat{\nu_l} = \sum^K_{k=1} \nu_{l,k}


0 \leq (\Delta h_{l}^+ + \Delta h_{l}^-) \perp \eta_l \geq 0


\pi^{\lambda}_l = \hat{\lambda}_{l-1} \odot \hat{\lambda_l}, \quad \pi^{\nu}_l = \hat{\nu}_{l-1} \odot \hat{\nu_l}


0=N_0\subset N_1\subset \cdots \subset N_k=N


V^{I_{v}^{(p)}} \otimes_R S \rightarrow (V \otimes_R S)^{I_{v}^{(p)}}


\cdots \rightarrow 0 \rightarrow V^{I_{v}^{(p)}} \xrightarrow{t-1} V^{I_{v}^{(p)}} \rightarrow 0 \rightarrow \cdots


0 \longrightarrow V^{I_v^{(p)}}\otimes_{R} S \longrightarrow V \otimes_{R} S \longrightarrow V/V^{I_v^{(p)}} \otimes_{R} S \longrightarrow 0