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| <p> $\tilde{z}^{i}_{t},{z}^{i}_{t}$ </p> |
| <p> $\mathcal{L}_{rec}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-\hat% |
| {Z}^{i}(\mathbf{r})\right\|^{2}$ </p> |
| <p> $\hat{Z^{i}}(\mathbf{r})$ </p> |
| <p> $\mathcal{L}_{Mrec}$ </p> |
| <p> $\mathcal{L}_{ref}=\sum_{\mathbf{r}\in\mathcal{R}}\left\|Z^{i}(\mathbf{r})-% |
| \tilde{Z}^{i}(\mathbf{r})\right\|^{2}\hskip 2.84544pt,\textrm{where}\hskip 2.8% |
| 4544pt\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i})$ </p> |
| <p> $\hat{Z}^{i}(\mathbf{r})=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\gamma(t))% |
| \mathbf{f}_{z}(\mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544% |
| ptT(t)=\textrm{exp}\left(\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$ </p> |
| <p> $t\mathbf{d}$ </p> |
| <p> $\nabla\mathcal{L}_{MDDS}$ </p> |
| <p> $\nabla_{\theta}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta}\mathcal{L}_{\mathrm{% |
| SDS}}(\mathbf{z},y_{src})-\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\hat{% |
| \mathbf{z}},y_{trg}),$ </p> |
| <p> $\lambda_{im}$ </p> |
| <p> $\mathcal{L}_{rtot}=\lambda_{rec}\mathcal{L}_{rec}+\lambda_{ref}\mathcal{L}_{ref}$ </p> |
| <p> $\lambda_{ref}$ </p> |
| <p> $y_{src}$ </p> |
| <p> $\mathcal{L}_{MDDS}$ </p> |
| <p> $\tilde{{Z}}^{i}$ </p> |
| <p> ${F_{\phi}(\cdot)}$ </p> |
| <p> $\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{DDS}}=\nabla_{\theta,\phi}\mathcal{L}% |
| _{\mathrm{SDS}}(\mathbf{z}^{i},y_{src})-\nabla_{\theta,\phi}\mathcal{L}_{% |
| \mathrm{SDS}}(\tilde{\mathbf{z}}^{i},y_{trg}).$ </p> |
| <p> $(\mathbf{f}_{z},\sigma)=F_{\theta}(\mathbf{\gamma}(\mathbf{x}),\mathbf{\gamma}% |
| (\mathbf{d}))$ </p> |
| <p> $\hat{C}(r)=\int_{t_{n}}^{t_{f}}T(t)\mathbf{\sigma}(\mathbf{r}(t))\mathbf{c}(% |
| \mathbf{r}(t),d)dt,\hskip 2.84544pt\textrm{where}\hskip 2.84544ptT(t)=\textrm{% |
| exp}\left(-\int_{t_{n}}^{t}\sigma(\mathbf{r}(s))ds\right).$ </p> |
| <p> $\nabla_{\theta}\mathcal{L}_{\mathrm{SDS}}(\mathbf{z},y_{trg},\epsilon,t)=% |
| \omega(t)(\epsilon_{\psi}\left(\mathbf{z}_{\mathbf{t}},y_{trg},t\right)-% |
| \epsilon)\frac{\partial\mathbf{z}_{\mathbf{t}}}{\partial\theta}$ </p> |
| <p> $504\times 378$ </p> |
| <p> $\lambda_{rec}$ </p> |
| <p> $\mathcal{L}_{\mathrm{Mrec}}=\lambda_{im}\cdot\mathcal{M}\cdot\mathcal{L}_{% |
| \mathrm{rtot}}+\lambda_{om}\cdot(1-\mathcal{M})\cdot\mathcal{L}_{\mathrm{rtot}}.$ </p> |
| <p> $I=\{I^{i}\}_{i=1}^{N}$ </p> |
| <p> ${z}:=\{{z}^{i}\}_{i=1}^{N}$ </p> |
| <p> $\tilde{z}^{i}$ </p> |
| <p> ${z^{i}}=\mathcal{E}({{I^{i}}})\in\mathbb{R}^{64\times 64\times 4}$ </p> |
| <p> $y_{trg}$ </p> |
| <p> $\nabla_{\theta,\phi}\mathcal{L}_{\mathrm{MDDS}}=\mathcal{M}\cdot(\nabla_{% |
| \theta,\phi}\mathcal{L}_{\mathrm{DDS}}),$ </p> |
| <p> $[t_{near},t_{far}]$ </p> |
| <p> $\mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{MDDS}}+\mathcal{L}_{\mathrm{% |
| Mrec}}$ </p> |
| <p> $\lambda_{om}$ </p> |
| <p> $\tilde{z}^{i}={F}_{\phi}(\hat{z}^{i}).$ </p> |
| <p> $\mathcal{L}_{rtot}$ </p> |
| <p> $\mathbf{r}(t)=\mathbf{o}$ </p> |
| <p> $[T\cdot G,Q]$ </p> |
| <p> $\displaystyle+R_{\text{upright}}+R_{\text{heading}}$ </p> |
| <p> $205\pm 8$ </p> |
| <p> $10P$ </p> |
| <p> $\displaystyle+R_{\text{effort}}+R_{\text{act}}+R_{\text{dof}}$ </p> |
| <p> $P=100,T=50,W=10$ </p> |
| <p> $12+3\mathbb{A}+2\mathbb{F}$ </p> |
| <p> $\mathbb{A},\mathbb{F}$ </p> |
| <p> $\displaystyle+R_{\text{death}}\times\mathbf{1}_{\{\text{head\_height}\leq\text% |
| {termination\_height}\}}$ </p> |
| <p> $\displaystyle\ R_{\text{progress}}+R_{\text{alive}}\times\mathbf{1}_{\{\text{% |
| head\_height}\geq\text{termination\_height}\}}$ </p> |
| <p> $G=\lfloor(Q-P)/T\rfloor$ </p> |
| <p> $(\sum_{i=0}^{\left\lfloor\frac{m-1}{6}\right\rfloor}\frac{1}{2}\Phi(p^{m-6i}))+1$ </p> |
| <p> $v\in\mathbb{Z}_{p^{m}}$ </p> |
| <p> $\tau^{\prime}\circ\tau_{i}(E)=\tau^{\prime}(E)=E^{\prime},\ \forall\ i\in\{1,2% |
| ,3,...,t\}$ </p> |
| <p> $j(E_{1})\neq 0,1728$ </p> |
| <p> $\begin{split}\quad y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\end{split}$ </p> |
| <p> $a^{\frac{p-1}{2}}\equiv 1\mod p)$ </p> |
| <p> $N_{R}(\mathbb{Z}_{p})=\Phi(p^{2})$ </p> |
| <p> $N_{R}(\mathcal{R})$ </p> |
| <p> $\ \ \frac{\Phi(n^{4})}{|Aut(E)|}$ </p> |
| <p> $32505856$ </p> |
| <p> $E_{2}/\mathbb{Z}_{p^{m}}:y^{2}=x^{3}+\bar{a}x+\bar{b}$ </p> |
| <p> $\displaystyle b_{6}$ </p> |
| <p> $\Delta^{v}(p^{m})$ </p> |
| <p> $2q-4+6+4=2q+6$ </p> |
| <p> $E_{1}/\mathbb{F}_{q}:y^{2}=x^{3}+ax+b$ </p> |
| <p> $|Aut(E)|=4$ </p> |
| <p> $\{u^{\prime},\alpha u^{\prime},\alpha^{2}u^{\prime},-u^{\prime},-\alpha u^{% |
| \prime},-\alpha^{2}u^{\prime}\}$ </p> |
| <p> $\{6,19\}$ </p> |
| <p> $\begin{split}C_{R}(\mathbb{Z}_{n})=\sum_{\mathbb{E}_{k}}C^{(k)}_{R}(\mathbb{Z}% |
| _{n})\end{split}$ </p> |
| <p> $34091302912$ </p> |
| <p> $15400$ </p> |
| <p> $(3^{-1})^{3}a^{3}\equiv-(2^{-1})^{2}b^{2}\ \mod p$ </p> |
| <p> $\mathbb{Z}_{2^{m}}$ </p> |
| <p> $k_{2}=4$ </p> |
| <p> $6\nmid char(\mathbb{Z}_{n})$ </p> |
| <p> $\displaystyle(E)=c_{4}^{3}/\Delta$ </p> |
| <p> $\displaystyle b_{8}$ </p> |
| <p> $3.5115653\times 10^{13}$ </p> |
| <p> $\frac{q-1}{|Aut(E)|}$ </p> |
| <p> $\{11,14\}$ </p> |
| <p> $2p^{m}-4$ </p> |
| <p> $28672$ </p> |
| <p> $P(1,1)$ </p> |
| <p> $N_{q}=2q+3+\left(\frac{-4}{q}\right)+2\left(\frac{-3}{q}\right)$ </p> |
| <p> $N_{G}(\mathbb{Z}_{n})=\Phi(n^{5})$ </p> |
| <p> $x=X/Z,y=Y/Z$ </p> |
| <p> $Aut(E^{\prime})$ </p> |
| <p> $y^{2}=x^{3}+3x$ </p> |
| <p> $a=0\;\text{and}\;b\neq 0$ </p> |
| <p> $|Aut(E)|=6$ </p> |
| <p> $\tau^{\prime}\neq\tau_{i}$ </p> |
| <p> $2p^{i-1}$ </p> |
| <p> $|Aut(E_{1})|=6$ </p> |
| <p> $q\equiv 1\mod k$ </p> |
| <p> $E/\mathbb{F}_{q}$ </p> |
| <p> $\{2,23\}$ </p> |
| <p> $|Aut(\mathbb{E}_{k})|\;\text{over}\;\mathbb{Z}_{n}=\prod\limits_{i=1}^{l}|Aut(% |
| \mathbb{E}_{k_{i}})|\;\text{over}\;\mathbb{Z}_{p_{i}}$ </p> |
| <p> $\mathbb{F}_{2^{m}},1\leq m\leq 10$ </p> |
| <p> $F(X,Y,Z)=Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}-X^{3}-a_{2}X^{2}Z-a_{4}XZ^{2}-a_{6}Z^{3}=0$ </p> |
| <p> $N^{\prime\prime}_{R}(\mathbb{Z}_{n})$ </p> |
| <p> $y^{2}=x^{3}+4x+2$ </p> |
| <p> $\mathbb{Z}_{p^{m}}^{*}$ </p> |
| <p> $Aut(E)=\{\tau_{1},\tau_{2},...,\tau_{t}\}$ </p> |
| <p> $\mathbb{Z}_{7^{m}}$ </p> |
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