paper_id stringclasses 7
values | pix2tex stringlengths 29 1.18k | 1.5B stringlengths 5 463 | 7B stringlengths 30 552 | gt stringlengths 5 470 |
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2512.05089v1 | \alpha\implies\beta(s,\theta), | \begin{equation}
\alpha \implies \beta(s, \theta),
\end{equation} | \alpha\implies\beta(s,\theta), | \[
x = f(s,\theta),
\] |
2512.05089v1 | {\mathcal A}{\bar{u}}\,=\,\{\,f(s,\theta{)}:s\in\mathcal{G},\;\theta\in\Theta\,\}\,\subset\,C^{0}([0,T]). | \[
\mathcal{A} \bar{u} \, = \, \{ f(s, \theta) : s \in \mathcal{G}, \, \theta \in \Theta \} \, \subset \, C^0([0, T]).
\] | \begin{equation}
\mathcal{A} \bar{u} = \{ f(s,\theta): s \in \mathcal{G}, \theta \in \Theta \} \subset C^0([0,T]).
\end{equation} | \[
\mathcal{M} = \{\, f(s,\theta) : s\in\mathcal{S},\;\theta \in \Theta \,\}
\subset C^0([0,T]).
\] |
2512.05089v1 | r:=\operatorname{sup}_{x\in{\mathcal{N}}}\left||x-x_{0}|\right|_{\infty} | \[
r := \sup_{x \in \mathcal{N}} \norm{x - x_0}_\infty
\] | \begin{equation}
r := \sup_{x \in \mathcal{N}} \norm{x - x_0}_\infty
\end{equation} | \[
r := \sup_{x\in\mathcal{M}} \|x-x_0\|_\infty
\] |
2512.05089v1 | {\mathcal N}\leftarrow B_{\infty}(x_{0},r)=\{x\in C^{0}:||x-x_{0}||\infty\leq r\}. | \[
\mathcal{N} \leftarrow B_\infty(x_0, r) = \{ x \in C^0 : \|x - x_0\|_\infty \leq r \}.
\] | \begin{equation}
\mathcal{N} \leftarrow B_{\infty}(x_0, r) = \{x \in C^0: \norm{x - x_0}_\infty \leq r\}.
\end{equation} | \[
\mathcal{M} \subset B_\infty(x_0,r)
= \{\,x \in C^0 : \|x-x_0\|_\infty \le r \,\}.
\] |
2512.05089v1 | \Phi:C^{0}([0,T])\longrightarrow\mathbb{R}, | \[
\Phi: C^0([0,T]) \to \mathbb{R},
\] | $$\Phi: C^0([0,T]) \to \mathbb{R},$$ | \[
\Phi : C^0([0,T]) \to \mathbb{R},
\] |
2512.05089v1 | r\implies\operatorname{sup}_{x\in{\mathcal{N}}}d_{H}(\{x\},\{x_{0}\})\Longrightarrow\operatorname*{sup}_{x\in{\mathcal{N}}}\|x-x_{0}\|_{\infty}. | \[
\operatorname{sup}_{x\in\mathcal{N}}d_{H}(\{x\},\{x_0\})\implies \operatorname*{sup}_{x\in\mathcal{N}}\|x-x_0\|_{\infty}.
\] | \begin{equation}
r = \sup_{x \in \mathcal{N}} d_H(\{x\}, \{x_0\}) = \sup_{x \in \mathcal{N}} \|x - x_0\|_{\infty}.
\end{equation} | \[
r = \sup_{x\in\mathcal{M}} d_H(\{x\}, \{x_0\})
= \sup_{x\in\mathcal{M}} \|x - x_0\|_\infty.
\] |
2512.05089v1 | \hat{r}_{n}\,\longrightarrow\,{\bf\Pi}_{{\bf T}\setminus\hat{\imath}\leq n}\,\left|\left|\right\rangle\Big(s_{i},\,\theta_{i}\right)\,-\,\mathcal{U}_{0}\,\right||\infty\,. | \begin{equation}
\hat{r}_n \, \longrightarrow \, \mathcal{P}_{\mathcal{T} \setminus \hat{\imath} \leq n} \left\lvert \left\lvert \hat{r}_n \right\rangle \Big(s_i, \theta_i \right) - \mathcal{U}_0 \right\rangle \infty \,.
\end{equation} | \begin{equation}
\hat{r}_n \,\longrightarrow\, \mathcal{P}_{\mathcal{T} \setminus \hat{\mathcal{I}} \leq n} \,\Big\| \big(s_i, \theta_i\big) - \mathcal{U}_0 \Big\|_{\infty} \,.
\end{equation} | \[
\hat{r}_n = \max_{1\le i \le n} \|f(s_i,\theta_i) - x_0\|_\infty.
\] |
2512.05089v1 | \left.\begin{array}{c}{{\mathrm{SUU}\nonumber}}\\ {{\left.x{\not{D}}q\right|}}\end{array}\left|\Phi\left(\mathcal{D}\big(\mathcal{D}\big)\right.\longrightarrow\left.\mathcal{N}_{\mathcal{E}}\left(\mathcal{D}\big)\right|\ <\mathcal{E}. | \left.\begin{array}{c}{{\mathrm{SUU}\nonumber}}\\ {{\left.x{\not{D}}q\right|}}\end{array}\left|\Phi\left(\mathcal{D}\big(\mathcal{D}\big)\right.\longrightarrow\left.\mathcal{N}_{\mathcal{E}}\left(\mathcal{D}\big)\right|\ <\mathcal{E}. | \left.\begin{array}{c}{{\mathrm{SUU}\nonumber}}\\ {{\left.x{\not{D}}q\right|}}\end{array}\left|\Phi\left(\mathcal{D}\big(\mathcal{D}\big)\right.\longrightarrow\left.\mathcal{N}_{\mathcal{E}}\left(\mathcal{D}\big)\right|\ <\mathcal{E}. | \\sup_{x \\in \\mathcal{M}} |\\Phi(x) - N_\\varepsilon(x)| < \\varepsilon |
2512.05089v1 | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{1\underline{{{\mit}}}\underline{{{\star}}}\underline{{{\l}}}}_{n}\,\vert\vert\alpha_{i}\,-\,\boldsymbol{x_{0}}\vert\vert\infty. | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{1\underline{{{\mit}}}\underline{{{\star}}}\underline{{{\l}}}}_{n}\,\vert\vert\alpha_{i}\,-\,\boldsymbol{x_{0}}\vert\vert\infty. | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{1\underline{{{\mit}}}\underline{{{\star}}}\underline{{{\l}}}}_{n}\,\vert\vert\alpha_{i}\,-\,\boldsymbol{x_{0}}\vert\vert\infty. | \[
\hat{r}_n = \max_{1\le i \le n} \|f(s_i,\theta_i) - x_0\|_\infty.
\] |
2512.05089v1 | \gamma\implies\mathrm{{\bf~Sup}}\left|\right|f\!\left(s,\theta\!\right)-\mathcal{A}_{0}\right|\right|\infty. | \[
\gamma \implies \mathrm{Sup} \left\| f(s, \theta) - \mathcal{A}_0 \right\|_{L^\infty}.
\] | \begin{equation}
\gamma \implies \sup_{\theta} \norm{f(s, \theta) - \mathcal{A}_0} \leq \infty.
\end{equation} | \[
r = \sup_{s,\theta} \| f(s,\theta) - x_0 \|_\infty.
\] |
2512.05089v1 | d_{H}(\{x\},\backslash W)<\varepsilon, | $$d_H(\{x\}, \backslash W) < \varepsilon,$$ | \begin{equation}
d_H(\{x\},\mathcal{W}) < \varepsilon,
\end{equation} | \[
d_H(\{x\},\mathcal{M}) = \inf_{y\in\mathcal{M}} \|x-y\|_\infty.
\] |
2512.05089v1 | \hat{r}_{n}\,\longrightarrow\,{\bf m}_{\bf T}\hat{\Sigma}_{n}\,\vert\vert\mathcal{T}_{i}\stackrel{}{\cal Q}_{0}\vert\vert\infty | \hat{r}_{n}\,\longrightarrow\,{\bf m}_{\bf T}\hat{\Sigma}_{n}\,\vert\vert\mathcal{T}_{i}\stackrel{}{\cal Q}_{0}\vert\vert\infty | \hat{r}_{n}\,\longrightarrow\,{\bf m}_{\bf T}\hat{\Sigma}_{n}\,\vert\vert\mathcal{T}_{i}\stackrel{}{\cal Q}_{0}\vert\vert\infty | \[\n\\|f(s_{i_k},\\theta_{i_k}) - x_0\\|_\\infty \\to \\|x^\\star - x_0\\|_\\infty = r.\n\\] |
2512.05089v1 | r=\operatorname*{sup}_{x\in{\mathcal{N}}}\left\|x-x_{0}\right\|_{\infty}, | \[
r = \sup_{x \in \mathcal{N}} \norm{x - x_0}_\infty,
\] | \begin{equation}
r = \sup_{x \in \mathcal{N}} \norm{x - x_0}_\infty,
\end{equation} | \[
r = \sup_{x \in \mathcal{M}} \|x - x_0\|_\infty,
\] |
2512.05089v1 | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{\l}\underline{{{\l}}}_{\l}\downarrow\slash\!\L_{i}\,-\,\boldsymbol{x}_{0}\Vert\cfrac{\cfrac{}{\L}_{0}.} | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{\l}\underline{{{\l}}}_{\l}\downarrow\slash\!\L_{i}\,-\,\boldsymbol{x}_{0}\Vert\cfrac{\cfrac{}{\L}_{0}.} | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{\l}\underline{{{\l}}}_{\l}\downarrow\slash\!\L_{i}\,-\,\boldsymbol{x}_{0}\Vert\cfrac{\cfrac{}{\L}_{0}.} | \[
\hat{r}_n = \max_{1 \le i \le n} \| x_i - x_0 \|_\infty.
\] |
2512.05089v1 | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{1\underline{{{\bf x}}}i\underline{{{\bf x}}}}\,{\left|{\vert}x_{i}\,-\,x_{0}\vert}{\vert}\right|_{\infty}, | \hat{r}_{n}\,\underline{{{\l}}}_{n}\,\underline{{{\l}}}_{1\underline{{{\bf x}}}i\underline{{{\bf x}}}}\,{\left|{\vert}x_{i}\,-\,x_{0}\vert}{\vert}\right|_{\infty}, | \begin{equation}
\hat{r}_n \leq \underline{{{\l}}}_{n} \underline{{{\l}}}_{1\underline{{{\bf x}}}i\underline{{{\bf x}}}}\,{\left|{\vert}x_{i}\,-\,x_{0}\vert}{\vert}\right|_{\infty},
\end{equation} | \[
\hat r_n = \max_{1 \le i \le n} \|x_i - x_0\|_\infty,
\] |
2512.05089v1 | \hat{r}_{n}\,=\,\mathrm{max}(\hat{r}_{n-1},d_{n}) | \begin{equation}
\hat{r}_n = \max(\hat{r}_{n-1}, d_n)
\end{equation} | \begin{equation}
\hat{r}_n = \max(\hat{r}_{n-1}, d_n)
\end{equation} | \[
\hat r_n = \max(\hat r_{n-1}, d_n)
\] |
2512.05089v1 | \mathrm{if}\ d_{n}>\hat{r}_{n-1}+\delta\Rightarrow\mathrm{new\\varialbility\detected} | \begin{equation}
\text{if } d_n > \hat{r}_{n-1} + \delta \Rightarrow \text{new variance detected}
\end{equation} | \mathrm{if}\ d_{n}>\hat{r}_{n-1}+\delta\Rightarrow\mathrm{new\\varialbility\detected} | \[\n\\text{if } d_n > \\hat r_{n-1} + \\delta \\Rightarrow \\text{new variability detected}\n\\] |
2512.05089v1 | d_{n}\ =||x_{n}-x_{0}||_{\infty} | \begin{equation}
d_n = \|x_n - x_0\|_{\infty}
\end{equation} | $$d_n = \norm{x_n - x_0}_\infty$$ | \[
d_n = \|x_n - x_0\|_\infty
\] |
2512.05089v1 | d_{H}(X_{n},X_{n/2}),~~~~~~~r_{\mathrm{max}}(X_{n}),~~~~~~~\bar{r}(X_{n}),~~~~~~~V_{\mathrm{bbox}}(X_{n}), | $$d_H(X_n, X_{n/2}),\quad r_{\mathrm{max}}(X_n),\quad \bar{r}(X_n),\quad V_{\mathrm{sup}}(X_n),$$ | d_{H}(X_{n},X_{n/2}),~~~~~~~~~~~r_{\mathrm{max}}(X_{n}),~~~~~~~~~~~\bar{r}(X_{n}),~~~~~~~~~~~V_{\mathrm{bbox}}(X_{n}), | \[
d_H(X_n, X_{n/2}), \qquad
r_{\max}(X_n), \qquad
\bar{r}(X_n), \qquad
V_{\mathrm{bbox}}(X_n),
\] |
2512.05089v1 | d_{H}(X_{n},X_{n/2})\approx10^{-2},\qquad r_{\mathrm{max}}(X_{n})\approx\mathrm{constant},\qquad V_{\mathrm{bbox}}(X_{n})\approx\mathrm{constant}. | $$d_H(X_n, X_{n/2}) \approx 10^{-2}, \qquad r_{\mathrm{max}}(X_n) \approx \text{constant}, \qquad V_{\mathrm{sup}}(X_n) \approx \text{constant}.$$ | \begin{equation}
d_H(X_n, X_{n/2}) \approx 10^{-2}, \qquad r_{\text{max}}(X_n) \approx \text{constant}, \qquad V_{\text{bbox}}(X_n) \approx \text{constant}.
\end{equation} | \[
d_H(X_n, X_{n/2}) \approx 10^{-2}, \qquad
r_{\max}(X_n) \approx \text{constant}, \qquad
V_{\mathrm{bbox}}(X_n) \approx \text{constant}.
\] |
2512.05089v1 | (d_{H})=\left[\chi_{\mathrm{real}},\chi_{\mathrm{sim}}\right],\qquad\qquad d_{H}(\mathrm{x}_{\mathrm{sim}},\mathrm{X}_{\mathrm{real}}), | \begin{equation}
d_{H} = \left[ \chi_{\text{real}}, \chi_{\text{sim}} \right], \qquad \qquad d_{H}(\text{x}_{\text{sim}}, \text{X}_{\text{real}}),
\end{equation} | \begin{equation}
d_H(\mathbf{x}_{\mathrm{sim}},\mathbf{x}_{\mathrm{real}}) = \left[\chi_{\mathrm{real}},\chi_{\mathrm{sim}}\right],
\end{equation} | \[
d_H(X_{\mathrm{real}}, X_{\mathrm{sim}}),
\qquad
d_H(X_{\mathrm{sim}}, X_{\mathrm{real}}),
\] |
2512.05089v1 | d_{H}(X_{n}^{\mathrm{sim}},X_{n/2}^{\mathrm{sim}})\approx10^{-2},\qquad r_{\mathrm{max}}(X_{n}^{\mathrm{sim}})\approx\mathrm{constant}. | $$d_H(X_n^{\text{sim}}, X_{n/2}^{\text{sim}}) \approx 10^{-2}, \qquad r_{\text{max}}(X_n^{\text{sim}}) \approx \text{constant}.$$ | \begin{equation}
d_H(X_n^{\text{sim}}, X_{n/2}^{\text{sim}}) \approx 10^{-2}, \qquad r_{\text{max}}(X_n^{\text{sim}}) \approx \text{constant}.
\end{equation} | \[
d_H(X^{\mathrm{sim}}_n, X^{\mathrm{sim}}_{n/2}) \approx 10^{-2},
\qquad r_{\max}(X^{\mathrm{sim}}_n) \approx \text{constant}.
\] |
2512.05089v1 | d_{H}({\mathrm{real}}\to{\mathrm{sim}})\in[0.03,\,0.07],\qquad d_{H}({\mathrm{sim}}\to{\mathrm{real}})\in[0.07,\,0.1]]. | $$d_H(\mathrm{real} \to \mathrm{sim}) \in [0.03, 0.07], \qquad d_H(\mathrm{sim} \to \mathrm{real}) \in [0.07, 0.1].$$ | d_{H}({\mathrm{real}}\to{\mathrm{sim}})\in[0.03,\,0.07],\qquad d_{H}({\mathrm{sim}}\to{\mathrm{real}})\in[0.07,\,0.1]]. | \[d_H(X_{\mathrm{real}}, X_{\mathrm{sim}}), \qquadd_H(X_{\mathrm{sim}}, X_{\mathrm{real}}),\] |
2512.05089v1 | r\,\underline{{{\begin{array}{c}{{\leq}}}\end{array}}}\operatorname*{sup}_{x\in\mathcal{N}}\left|\left|\mathcal{x}\,-\,\mathcal{X}_{0}\right|\right|_{\infty}\,<\,\infty. | \[
r \leq \sup_{x \in \mathcal{N}} \|\mathcal{x} - \mathcal{X}_0\|_\infty.
\] | \begin{equation}
\lim_{n\to\infty}\sup_{x\in\mathcal{N}}\left\|\mathbf{x}-\mathbf{x}_0\right\|_{\infty}=0.
\end{equation} | \[r = \sup_{s, heta} \| f(s, heta) - x_0 \|_\infty.] |
2512.05089v1 | \|x-x_{0}\|_{\infty}\leq\|x\|_{\infty}+\|x_{0}\|_{\infty}\leq2R. | \[
\|x - x_0\|_{\infty} \leq \|x\|_{\infty} + \|x_0\|_{\infty} \leq 2R.
\] | \[
\|x - x_0\|_{\infty} \leq \|x\|_{\infty} + \|x_0\|_{\infty} \leq 2R.
\] | \[\|x - x_0\|_\infty \le \|x\|_\infty + \|x_0\|_\infty \le 2R.] |
2512.05089v1 | |t_{1}-t_{2}|<\delta\quad\Rightarrow\quad|f(s,\theta)(t_{1})-f(s,\theta)(t_{2})|<\varepsilon. | $$|t_1-t_2|<\delta\quad\Rightarrow\quad|f(s,\theta)(t_1)-f(s,\theta)(t_2)|<\varepsilon.$$ | \begin{equation}
|t_1-t_2| < \delta \quad \Rightarrow \quad |f(s,\theta)(t_1) - f(s,\theta)(t_2)| < \varepsilon.
\end{equation} | \[
|t_1 - t_2| < \delta
\quad\Rightarrow\quad
|f(s,\theta)(t_1) - f(s,\theta)(t_2)| < \varepsilon.
\] |
2512.05089v1 | \left|\right|f{\big(}s_{i_{k}},\theta_{i_{k}}{\big)}\longrightarrow x_{0}\left|\right|\left|\left|\infty\right.\longrightarrow\left|\left|x^{\star}-x_{0}\right|\right|_{\infty}\ =\,r. | \[
\|f(s_{i_k}, \theta_{i_k}) - x_0\|_{\infty} \leq r.
\] | \begin{equation}
\|f(s_{i_k}, \theta_{i_k}) - x_0\|_{\infty} \to r.
\end{equation} | \[\|f(s_{i_k}, heta_{i_k}) - x_0\|_\infty o \|x^\star - x_0\|_\infty = r.] |
2512.05089v1 | \gamma\implies\operatorname{sup}_{x\in{\mathcal{N}}}\left||x-\operatorname{\mathcal{A}}_{0}|\right|_{\infty} | \[
\gamma \implies \sup_{x \in \mathcal{N}} \norm{x - \mathcal{A}_0}_\infty
\] | \begin{equation}
\gamma = \sup_{x \in \mathcal{N}} \norm{x - \mathcal{A}_0}_\infty
\end{equation} | \[
r = \sup_{x\in\mathcal{M}} \|x - x_0\|_\infty < \infty.
\] |
2512.05089v1 | \left|\left|x-y\right|\right|_{\infty}<\delta\pmod^{}\ \ \ \ \Rightarrow\ \ \ \left|\Phi(x)-\Phi(y)\right|<\varepsilon | \[
\|\x-\y\|_\infty < \delta \quad \implies \quad \|\Phi(\x)-\Phi(\y)\|<\varepsilon
\] | \begin{equation}
\|x-y\|_{\infty} < \delta \quad \Rightarrow \quad |\Phi(x) - \Phi(y)| < \varepsilon
\end{equation} | \[\n\\|x - y\\|_\\infty < \\delta \n\\quad\\Rightarrow\\quad\n|\\Phi(x) - \\Phi(y)| < \\varepsilon\n\\] |
2512.05089v1 | \int\left(S_{i_{k}}\,,\,\theta_{i_{k}}\right)\longrightarrow\mathcal{D}^{\star}. | \begin{equation}
\int \left(S_{i_k}, \theta_{i_k}\right) \longrightarrow \mathcal{D}^{\star}.
\end{equation} | \begin{equation}
\int (S_{i_k}, \theta_{i_k}) \to \mathcal{D}^{\star}.
\end{equation} | \[
f(s_{i_k}, heta_{i_k}) o x^\star.
\] |
2512.05089v1 | \hat{\hat{r}}_{n}\ \longrightarrow\ \mathrm{\bf~{\bf~T}}_{\textstyle{\sum\hat{\Sigma}}\hat{\imath}\leq n}\left|\right|\J\hat{f}\left({\cal S}_{i}\,,\,\theta_{i}\right)\ -\ {\cal T}_{0}\left|\right|_{\textstyle\infty}. | \hat{\hat{r}}_{n}\ \longrightarrow\ \mathrm{\bf~{\bf~T}}_{\textstyle{\sum\hat{\Sigma}}\hat{\imath}\leq n}\left|\right|\J\hat{f}\left({\cal S}_{i}\,,\,\theta_{i}\right)\ -\ {\cal T}_{0}\left|\right|_{\textstyle\infty}. | \begin{equation}
\hat{\hat{r}}_{n}\ \longrightarrow\ \mathrm{\bf~{\bf~T}}_{\textstyle{\sum\hat{\Sigma}}\hat{\imath}\leq n}\left|\right|\J\hat{f}\left({\cal S}_{i}\,,\,\theta_{i}\right)\ -\ {\cal T}_{0}\left|\right|_{\textstyle\infty}.
\end{equation} | \[\hat{r}_n = \max_{1\le i \le n} \|f(s_i, heta_i) - x_0\|_\infty.\] |
2512.05089v1 | \|x-y\|_{\infty}<\varepsilon, | \[
\|x-y\|_\infty < \varepsilon,
\] | \[
\|x-y\|_{\infty} < \varepsilon,
\] | \[\|x - y\|_\infty < \varepsilon,\] |
2512.05089v1 | d_{H}(\{x\},\mathcal{N})=\operatorname*{inf}_{y\in\mathcal{N}}\|x-y\|_{\infty}. | $$d_H(\{x\}, \mathcal{N}) = \inf_{y \in \mathcal{N}} \|x-y\|_\infty.$$ | \begin{equation}
d_H(\{x\}, \mathcal{N}) = \inf_{y \in \mathcal{N}} \norm{x-y}_\infty.
\end{equation} | \[\nd_H(\\{x\\},\\mathcal{M}) = \\inf_{y\\in\\mathcal{M}} \\|x-y\\|_\\infty.\n\\] |
2512.05093v1 | {\cal L}_{\mathrm{line}}^{\prime}=(3.25\times10^{7})\,I_{\mathrm{obs}}\,D_{L}^{2}\,(1+z)^{-3}\,\nu_{\mathrm{obs}}^{-2}[\mathrm{K\,km\,s^{-1}p c^{-2}}], | \begin{equation}
L_{\text{line}}' = (3.25 \times 10^7) \, I_{\text{obs}} \, D_L^2 \, (1+z)^{-3} \, \nu_{\text{obs}}^{-2} \left[ \text{K~km~s}^{-1} \text{pc}^{-2} \right],
\end{equation} | \begin{equation}
\mathcal{L}_{\text{line}}' = (3.25 \times 10^7) \, I_{\text{obs}} \, D_L^2 \, (1+z)^{-3} \, \nu_{\text{obs}}^{-2} [\text{K \, km \, s^{-1} pc^{-2}}],
\end{equation} | \begin{equation}
L^{'}_{\rm line} = (3.25 \times 10^7)\,I_{\rm obs}\,D^{2}_{L}\,(1+\textit{z})^{-3}\, \nu_{\rm obs}^{-2} [\rm K \, km\, s^{-1} pc^{-2}],
\end{equation} |
2512.05093v1 | {\cal L}_{\mathrm{line}}=(1.04\times10^{-3})\,I_{\mathrm{obs}}\,\nu_{\mathrm{rest}}\,D_{L}^{2}\,(1+z)^{-1}[{\cal L}_{\odot}], | \begin{equation}
L_{\text{line}} = (1.04\times10^{-3})\,I_{\text{obs}}\,\nu_{\text{rest}}\,D_L^2\,(1+z)^{-1}[L_{\odot}],
\end{equation} | \begin{equation}
\mathcal{L}_{\text{line}} = (1.04 \times 10^{-3})\, I_{\text{obs}}\, \nu_{\text{rest}}\, D_L^2\, (1+z)^{-1}[\mathcal{L}_{\odot}],
\end{equation} | \begin{equation}
L_{\rm line} = (1.04 \times 10^{-3})\,I_{\rm obs}\, \nu_{\rm rest}\,D^{2}_{L}\,(1+\textit{z})^{-1} [L_{\odot}],
\end{equation} |
2512.05094v1 | \beta_{t}^{\mathrm{Sym}}\underline{{{\pi}}}\frac{\tau(T_{a}(a_{t})|T_{s}(s_{t}))}{\pi_{\mathrm{old}}(a_{t}|s_{t})} | \begin{equation}
\beta_t^{\text{Sym}} \underline{{\pi}} \frac{\tau(T_a(a_t)|T_s(s_t))}{\pi_{\text{old}}(a_t|s_t)}
\end{equation} | \begin{equation}
\beta_t^{\mathrm{Sym}} = \frac{\tau(T_a(a_t) | T_s(s_t))}{\pi_{\mathrm{old}}(a_t | s_t)}
\end{equation} | \begin{equation}
\beta_t^{\text{sym}} = \frac{\pi(T_a(a_t) | T_s(s_t))}{\pi_{\text{old}}(a_t | s_t)}
\end{equation} |
2512.05094v1 | {\cal{R}}_{t}^{\mathrm{weighted}}=\exp\left(-\sum_{j=1}^{n}w_{j}||p_{t,j}-p_{t,j}^{\mathrm{goal}}||_{2}^{2}/\sigma^{2}\right) | \begin{equation}
\mathcal{R}_t^{\text{Weighted}} = \exp\left(-\sum_{j=1}^n w_j \|p_{t,j} - p_{t,j}^{\text{goal}}\|_2^2/\sigma^2\right)
\end{equation} | \begin{equation}
\mathcal{R}^{\text{weighted}}_t = \exp\left(-\sum_{j=1}^n w_j \|\mathbf{p}_{t,j} - \mathbf{p}^{\text{goal}}_{t,j}\|_2^2 / \sigma^2\right)
\end{equation} | \begin{equation}
\mathcal{R}_t^{\text{weighted}} = \exp\left(- \sum_{j=1}^{n} w_j \| p_{t,j} - p_{t,j}^{\text{goal}} \|_2^2 / \sigma^2 \right)
\end{equation} |
2512.05094v1 | \mathcal{L}_{\mathrm{PPO-SYM}}(\mathbf{\Lambda}\mathbf{\Lambda})=\mathcal{L}_{\mathrm{PPO}}(\tau)+\lambda_{\mathrm{SYM}}\mathcal{L}_{\mathrm{SYM}}(\tau) | \begin{equation}
\mathcal{L}_{\mathrm{PPO-SSY}}(\mathbf{\Lambda}\mathbf{\Lambda}) = \mathcal{L}_{\mathrm{PPO}}(\tau) + \lambda_{\mathrm{PPO}} \mathcal{L}_{\mathrm{PPO}}(\tau)
\end{equation} | \begin{equation}
\mathcal{L}_{\mathrm{PPO-SYM}}(\mathbf{\Lambda}\mathbf{\Lambda}) = \mathcal{L}_{\mathrm{PPO}}(\tau) + \lambda_{\mathrm{SYM}} \mathcal{L}_{\mathrm{SYM}}(\tau)
\end{equation} | \begin{equation}
\mathcal{L}_{\text{PPO-SYM}}(\pi) = \mathcal{L}_{\text{PPO}}(\pi) + \lambda _{\text{SYM}} \mathcal{L}_{\text{SYM}}(\pi)
\end{equation} |
2512.05094v1 | T_{t}\,=\,K_{p}(q_{t}^{\mathrm{des}}-q_{t})\,-\,K_{d}\dot{q}_{t} | \begin{equation}
T_t = K_p (q_t^{\text{des}} - q_t) - K_d \dot{q}_t
\end{equation} | \begin{equation}
T_t = K_p (q_t^{\mathrm{des}} - q_t) - K_d \dot{q}_t
\end{equation} | \begin{equation}
T_t = K_p (q_t^{\text{des}} - q_t) - K_d \dot{q}_t
\end{equation} |
2512.05094v1 | q_{t}^{\mathrm{des}}\ =\,q^{\mathrm{default}}+c\cdot\mathrm{clip}(a_{t},-a^{\mathrm{clip}},a^{\mathrm{clip}}) | \begin{equation}
q_t^{\text{des}} = q^{\text{default}} + c \cdot \text{clip}(a_t, -a^{\text{clip}}, a^{\text{clip}})
\end{equation} | \begin{equation}
q_t^{\text{des}} = q^{\text{default}} + c \cdot \text{clip}(a_t, -a^{\text{clip}}, a^{\text{clip}})
\end{equation} | \begin{equation}
q_t^{\text{des}} = q^{\text{default}} + c \cdot \text{clip}(a_t, -a^{\text{clip}}, a^{\text{clip}})
\end{equation} |
2512.05096v1 | F_{\mathrm{mw}}=\frac{1}{4}\big(1+e^{-\tau/T_{1,\mathrm{t}}}+\Sigma e^{-\tau/2T_{1,\mathrm{t}}}e^{-\tau/T_{2,\mathrm{s}}}\big). | \begin{equation}
F_{\text{w}} = \frac{1}{4} \big(1 + e^{-\tau/T_{1,\text{t}}} + \Sigma e^{-\tau/2T_{1,\text{t}}} e^{-\tau/T_{2,\text{s}}}\big).
\end{equation} | \begin{equation}
F_{\rm mw} = \frac{1}{4}\big(1+e^{-\tau/T_{1,\rm t}}+\Sigma e^{-\tau/2T_{1,\rm t}}e^{-\tau/T_{2,\rm s}}\big).
\end{equation} | \begin{equation}
F_{\text{mw}}=\frac{1}{4}(1+e^{-\tau/T_{1,\text{t}}}+2e^{-\tau/2T_{1,\text{t}}}e^{-\tau/T_{2,\text{s}}}).
\end{equation} |
2512.05096v1 | \delta I=\omega_{R}\sqrt{\frac{\hbar}{2Z}}, | \begin{equation}
\delta I = \omega_R \sqrt{\frac{\hbar}{2Z}},
\end{equation} | \begin{equation}
\delta I = \omega_R \sqrt{\frac{\hbar}{2Z}},
\end{equation} | \begin{equation}
\delta I=\omega_R\sqrt{\frac{\hbar}{2Z}}.
\end{equation} |
2512.05096v1 | P_{\mathrm{mw}}=1-e^{-\Gamma\tau}. | \begin{equation}
P_{\text{w}} = 1 - e^{-\Gamma \tau}.
\end{equation} | \begin{equation}
P_{\rm mw} = 1 - e^{-\Gamma \tau}.
\end{equation} | \begin{equation}
P_{\text{mw}} =1-e^{-\Gamma \tau}.
\end{equation} |
2512.05096v1 | \delta I^{2}=\frac{\hbar\omega_{R}}{2Z_{0}}\,\int_{-\infty}^{\infty}2\,\left|\chi_{I I}(\omega)\right|^{2}\frac{d\omega}{2\pi}, | \begin{equation}
\delta I^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^{\infty} 2 \, \left| \chi_{II}(\omega) \right|^2 \frac{d\omega}{2\pi},
\end{equation} | \begin{equation}
\delta I^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^{\infty} 2 \, \left| \chi_{II}(\omega) \right|^2 \frac{d\omega}{2\pi},
\end{equation} | \begin{equation}
{\delta I}^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^\infty 2\ |\chi_{II}(\omega)|^2 \frac{d\omega}{2 \pi},
\end{equation} |
2512.05096v1 | R_{\mathrm{herald}}=0.5\cdot\frac{C^{2}}{\left(1+C\right)^{2}}\cdot\frac{\left(1-e^{-\Gamma\tau}\right)}{\tau+T_{\mathrm{reset}}}. | \begin{equation}
R_{\text{herd}} = 0.5 \cdot \frac{C^2}{(1+C)^2} \cdot \frac{1 - e^{-\Gamma \tau}}{\tau + T_{\text{ erase}}}.
\end{equation} | \begin{equation}
R_{\text{herald}} = 0.5 \cdot \frac{C^2}{(1+C)^2} \cdot \frac{(1 - e^{-\Gamma \tau})}{\tau + T_{\text{ reset}}}.
\label{eq:Rherald}
\end{equation} | \begin{equation}
R_{\text{herald}} = 0.5 \cdot \frac{C^2}{(1+C)^2} \cdot \frac{(1-e^{-\Gamma \tau})}{\tau + T_{\text{reset}}}.
\end{equation} |
2512.05096v1 | {\cal L}_{K,\Xi}=\frac{R_{\odot}\hbar}{\pi\Delta}\coth\left(\frac{\Delta}{2k_{B}T}\right), | \begin{equation}
L_{K, \Lambda} = \frac{R_{\odot} \hbar}{\pi \Delta} \coth \left( \frac{\Delta}{2 k_B T} \right),
\end{equation} | \begin{equation}
\mathcal{L}_{K,\Lambda} = \frac{R_\odot \hbar}{\pi \Delta} \coth\left(\frac{\Delta}{2k_B T}\right),
\end{equation} | \begin{equation}
L_{K,\square} = \frac{R_{\square}\hbar}{\pi \Delta}
\coth\left(\frac{\Delta}{2k_B T}\right),
\end{equation} |
2512.05096v1 | \delta I^{2}=\left\langle I_{L}^{2}(0)\right\rangle=\int_{-\infty}^{\infty}\left|I_{L}(\omega)\right|^{2}\;\frac{d\omega}{2\pi} | \begin{equation}
\delta I^2 = \langle I_L^2(0) \rangle = \int_{-\infty}^{\infty} \left| I_L(\omega) \right|^2 \frac{d\omega}{2\pi}
\end{equation} | \begin{equation}
\delta I^2 = \langle I_L^2(0) \rangle = \int_{-\infty}^{\infty} |I_L(\omega)|^2 \frac{d\omega}{2\pi}
\end{equation} | \begin{equation}
{\delta I}^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^\infty 2\ |\chi_{II}(\omega)|^2 \frac{d\omega}{2 \pi}.\label{I_L equation}
\end{equation} |
2512.05096v1 | =2\int_{-\infty}^{\infty}\left|\chi_{I I}(\omega)\right|^{2}S_{++}(\omega)\frac{d\omega}{2\pi}, | \begin{equation}
\label{eq:chisq}
\chi_{II}(\omega) = 2 \int_{-\infty}^{\infty} \left|\chi_{II}(\omega)\right|^2 S_{ plus}(\omega) \frac{d\omega}{2\pi},
\end{equation} | \begin{equation}
= 2 \int_{-\infty}^{\infty} \left| \chi_{II}(\omega) \right|^2 S_{++}(\omega) \frac{d\omega}{2\pi},
\end{equation} | \begin{equation}
{\delta I}^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^\infty 2\left|\frac{I_L(\omega)}{I_\text{in}(\omega)}\right|^2 \left|1-S_{11}\right|^2 \frac{d\omega}{2 \pi}.
\end{equation} |
2512.05096v1 | \delta I^{2}=\frac{h\omega_{R}}{2Z_{0}}\,\int_{-\infty}^{\infty}2\,\vert\chi_{I I}(\omega)\vert^{2}\frac{d\omega}{2\pi}. | \begin{equation}
\delta I^2 = \frac{h\omega_R}{2Z_0} \int_{-\infty}^{\infty} 2 \, |\chi_{II}(\omega)|^2 \frac{d\omega}{2\pi}.
\end{equation} | \begin{equation}
\delta I^2 = \frac{h\omega_R}{2Z_0} \int_{-\infty}^\infty 2\, |\chi_{II}(\omega)|^2 \frac{d\omega}{2\pi}.
\label{eq:deltaI2}
\end{equation} | \begin{align}
{\delta I}^2&=\left\langle I_L^2(0) \right\rangle = \int_{-\infty}^\infty |I_L(\omega)|^2 ~ \frac{d\omega}{2 \pi} \\
&= 2 \int_{-\infty}^\infty |\chi_{II}(\omega)|^2 S_{++}(\omega) \frac{d\omega}{2 \pi},\label{WK theorem}
\end{align} |
2512.05096v1 | \delta I^{2}=\frac{\hbar\omega_{R}}{2Z_{0}}\,\int_{-\infty}^{\infty}2\left|\frac{I_{L}(\omega)}{I_{\mathrm{in}}(\omega)}\right|^{2}|1-S_{11}|^{2}\,\frac{d\omega}{2\pi}. | \begin{equation}
\delta I^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^{\infty} 2 \left| \frac{I_L(\omega)}{I_{\mathrm{in}}(\omega)} \right|^2 |1-S_{11}|^2 \, \frac{d\omega}{2\pi}.
\end{equation} | \begin{equation}
\delta I^2 = \frac{\hbar \omega_R}{2 Z_0} \int_{-\infty}^{\infty} 2 \left| \frac{I_L(\omega)}{I_{\rm in}(\omega)} \right|^2 |1 - S_{11}|^2 \frac{d \omega}{2 \pi}.
\label{eq:deltaI2}
\end{equation} | \begin{equation}
\delta I^2 = \frac{\hbar{\omega_R}}{Z_0}\cdot\frac{Z_0}{Z}\frac{\omega_R}{2}={\omega_R}^2\frac{\hbar}{2Z},
\end{equation} |
2512.05096v1 | Z_{k}={\frac{1}{i\omega C_{k}}}+{\frac{i\omega L}{1-\omega^{2}L C}}. | \begin{equation}
Z_k = \frac{1}{i\omega C_k} + \frac{i\omega L}{1-\omega^2 L C_k}.
\end{equation} | \begin{equation}
Z_k = \frac{1}{i\omega C_k} + \frac{i\omega L}{1 - \omega^2 L C}.
\end{equation} | \begin{equation}\n\\frac{I_L}{I_{\\text{in}}}=\\frac{Z_0}{Z_0+Z_k}\\frac{1}{1-\\omega^2 L C}.\n\\end{equation} |
2512.05096v1 | \frac{1}{Z_{R}}=\frac{1}{Z_{0}}+\frac{1}{Z_{k}}, | \begin{equation}
\frac{1}{Z_R} = \frac{1}{Z_0} + \frac{1}{Z_k},
\end{equation} | \begin{equation}
\frac{1}{Z_R} = \frac{1}{Z_0} + \frac{1}{Z_k},
\end{equation} | \begin{equation}
\frac{1}{Z_R} = \frac{1}{Z_0} + \frac{1}{Z_k},
\end{equation} |
2512.05096v1 | S_{11}=\frac{Z_{R}-Z_{0}}{Z_{R}+Z_{0}}\rightarrow1-{\cal S}_{11}=\frac{2Z_{0}}{Z_{R}+Z_{0}}. | \begin{equation}
S_{11} = \frac{Z_R - Z_0}{Z_R + Z_0} \rightarrow 1 - S_{11} = \frac{2Z_0}{Z_R + Z_0}.
\end{equation} | \begin{equation}
S_{11} = \frac{Z_R - Z_0}{Z_R + Z_0} \approx 1 - \tilde{S}_{11} = \frac{2 Z_0}{Z_R + Z_0}.
\end{equation} | \begin{equation}
S_{11} = \frac{Z_{R}-Z_0}{Z_{R}+Z_0} \rightarrow 1-S_{11} = \frac{2 Z_0}{Z_{R}+Z_0}.
\end{equation} |
2512.05096v1 | \chi_{I I}=\frac{I_{L}}{I_{\mathrm{in}}}\big(1-S_{11}\big). | \begin{equation}
\chi_{II} = \frac{I_L}{I_{\mathrm{in}}} \big(1-S_{11}\big).
\end{equation} | \begin{equation}
\chi_{II} = \frac{I_L}{I_{\text{in}}}\big(1 - S_{11}\big).
\end{equation} | \begin{equation}\label{current suscep}
\chi_{II} = \frac{I_L}{I_\text{in}}(1-S_{11}).
\end{equation} |
2512.05096v1 | S_{++}(\omega)=\frac{h\omega}{2Z_{0}} | \begin{equation}
S_{++}(\omega)=\frac{h\omega}{2Z_0}
\end{equation} | \begin{equation}
S_{++}(\omega) = \frac{h \omega}{2 Z_0}
\end{equation} | \begin{equation}\nS_{++}(\\omega)=\\frac{\\hbar \\omega}{2 Z_0}.\n\\end{equation} |
2512.05096v1 | {\cal L}_{\mathrm{in}}=(1-S_{11})I^{+}. | \begin{equation}
\mathcal{L}_{\mathrm{in}} = (1-S_{11})I^+.
\end{equation} | \begin{equation}
\mathcal{L}_{\mathrm{in}} = (1 - S_{11}) I^+.
\end{equation} | \begin{equation}\nI_{\\text{in}}=(1-S_{11})\\,I^{+}.\n\\end{equation} |
2512.05096v1 | \delta I^{2}={\frac{\hbar\omega_{R}}{Z_{0}}}\cdot{\frac{Z_{0}}{Z}}\,{\frac{\omega_{R}}{2}}=\omega_{R}{}^{2}{\frac{h}{2Z}}, | \begin{equation}
\delta I^2 = \frac{\hbar \omega_R}{Z_0} \cdot \frac{Z_0}{Z} \cdot \frac{\omega_R}{2} = \omega_R^2 \frac{h}{2Z},
\end{equation} | \begin{equation}
\delta I^2 = \frac{\hbar \omega_R}{Z_0} \cdot \frac{Z_0}{Z} \cdot \frac{\omega_R}{2} = \omega_R^2 \frac{\hbar}{2Z},
\end{equation} | \begin{equation}\n{\\delta I}^2=\\frac{\\hbar \\omega_R}{2 Z_0}\\int_{-\\infty}^{\\infty}2\\,|\\chi_{II}(\\omega)|^2\\frac{d\\omega}{2\\pi}.\n\\end{equation} |
2512.05096v1 | \chi_{I I}(\omega^{\prime})=\frac{2\omega^{\prime}C^{\prime}}{2i-\omega^{\prime}C^{\prime}-2i\omega^{\prime2}(1+L^{\prime}C^{\prime})+\omega^{\prime3}C^{\prime}}. | \begin{equation}
\chi_{II}(\omega') = \frac{2\omega' C'}{2i - \omega' C' - 2i\omega'^2(1 + L' C') + \omega'^3 C'}.
\end{equation} | \begin{equation}
\chi_{II}(\omega') = \frac{2\omega'C'}{2i - \omega'C' - 2i\omega'^2(1 + L'C') + \omega'^3C'}.
\end{equation} | \chi_{I I}(\omega^{\prime})=\frac{2\omega^{\prime}C^{\prime}}{2i-\omega^{\prime}C^{\prime}-2i\omega^{\prime2}(1+L^{\prime}C^{\prime})+\omega^{\prime3}C^{\prime}}. |
2512.05096v1 | Z^{\prime}=\frac{Z_{k}}{Z_{0}},\;\;\;\;\omega^{\prime}=\frac{\omega}{\omega_{R}},\;\;\;\;C^{\prime}=\omega_{R}Z_{0}C_{k},\;\;\;\;L^{\prime}=\frac{\omega_{R}L}{Z_{0}}, | \begin{equation}
Z' = \frac{Z_k}{Z_0}, \quad \omega' = \frac{\omega}{\omega_R}, \quad C' = \omega_R Z_0 C_k, \quad L' = \frac{\omega_R L}{Z_0},
\end{equation} | \begin{equation}
Z' = \frac{Z_k}{Z_0}, \quad \omega' = \frac{\omega}{\omega_R}, \quad C' = \omega_R Z_0 C_k, \quad L' = \frac{\omega_R L}{Z_0},
\end{equation} | \[\nZ' = \\frac{Z_k}{Z_0},\\quad\n\\omega' = \\frac{\\omega}{\\omega_R},\\quad\nC' = \\omega_R Z_0 C_k,\\quad\nL' = \\frac{\\omega_R L}{Z_0}.\n\\] |
2512.05096v1 | \begin{array}{c}{{Z^{\prime}=\frac{1}{i\omega^{\prime}C^{\prime}}+\frac{i\omega^{\prime}L^{\prime}}{1-S_{11}},}}\\ {{\frac{I_{L}}{1+Z^{\prime}}\frac{1}{1-\omega^{\prime2}}.}}\end{array} | \begin{equation}
\begin{split}
Z'=\frac{1}{i\omega'C'}+\frac{i\omega'D'}{1-S_{11}},\\
\frac{I_L}{1+Z'}\frac{1}{1-\omega'^2}.
\end{split}
\end{equation} | \begin{equation}
Z' = \frac{1}{i\omega'C'} + \frac{i\omega'L'}{1 - S_{11}},
\label{eq:Zp}
\end{equation} | \begin{equation}\nZ' = \\frac{1}{i\\omega' C'} + \\frac{i\\omega' L'}{1-\\omega'^2},\n\\label{eq:Zp}\n\\end{equation} |
2512.05096v1 | \frac{I_{L}}{I_{\mathrm{in}}}=\frac{Z_{0}}{Z_{0}+Z_{k}}\frac{1}{1-\omega^{2}L C}. | \begin{equation}
\frac{I_L}{I_{\mathrm{in}}} = \frac{Z_0}{Z_0+Z_k} \frac{1}{1-\omega^2 LC}.
\end{equation} | \begin{equation}
\frac{I_L}{I_{\text{in}}} = \frac{Z_0}{Z_0 + Z_k} \frac{1}{1 - \omega^2 L C}.
\end{equation} | \begin{equation}
\\frac{I_L}{I_{\\text{in}}} = \\frac{Z_0}{Z_0+Z_k}\\frac{1}{1-\\omega^2 LC}.
\\end{equation} |
2512.05096v1 | {L}=\frac{{\cal Z}}{\omega_{R}}=\frac{h}{2}\frac{\omega_{R}}{\delta{\cal I}^{2}}. | \begin{equation}
L = \frac{Z}{\omega_R} = \frac{h}{2} \frac{\omega_R}{\delta I^2}.
\end{equation} | \begin{equation}
L = \frac{\mathcal{Z}}{\omega_R} = \frac{h}{2} \frac{\omega_R}{\delta I^2}.
\end{equation} | \begin{equation}
L = \frac{\mathcal{Z}}{\omega_R} = \frac{\hbar}{2} \frac{\omega_R}{\delta I^2}.
\end{equation} |
2512.05096v1 | C=\frac{1}{\omega_{R}Z}=\frac{2\delta I^{2}}{\hbar\omega_{R}3}, | \begin{equation}
C = \frac{1}{\omega_R Z} = \frac{2\delta I^2}{\hbar \omega_R 3},
\end{equation} | \begin{equation}
C = \frac{1}{\omega_R Z} = \frac{2 \delta I^2}{\hbar \omega_R 3},
\end{equation} | \begin{equation}
C = \frac{1}{\omega_R Z} = \frac{2 \delta I^2}{\hbar \omega_R 3},
\end{equation} |
2512.05096v1 | \delta I=\omega_{R}\sqrt{\frac{\hbar}{2Z}}. | \begin{equation}
\delta I = \omega_R \sqrt{\frac{\hbar}{2Z}}.
\end{equation} | \begin{equation}
\delta I = \omega_R \sqrt{\frac{\hbar}{2Z}}.
\end{equation} | \begin{equation}\\label{eq:di_z}
\\delta I = \\omega_R \\sqrt{\\frac{\\hbar}{2 Z}},
\\end{equation} |
2512.05096v1 | \mathcal{I}_{m}=-\gamma_{e}\delta\mathbf{B}\cdot\left\langle0\right|\mathbf{B}\left|1\right\rangle, | \begin{equation}
\mathcal{I}_m = -\gamma_e \delta \mathbf{B} \cdot \langle 0 | \mathbf{B} | 1 \rangle,
\end{equation} | \begin{equation}
\mathcal{I}_m = -\gamma_e \delta \mathbf{B} \cdot \langle 0 | \mathbf{B} | 1 \rangle,
\end{equation} | \begin{equation}
g_m = -\gamma_e \delta\mathbf{B}\cdot\langle 0|\mathbf{S}|1\rangle,
\end{equation} |
2512.05096v1 | g_{o}(\vec{r})=\frac{\mu E}{\hbar}=\sqrt{\frac{\mu^{2}\omega_{o}}{2\epsilon_{0}\hbar V}}u(\vec{r}), | \begin{equation}
g_o(\vec{r}) = \frac{\mu E}{\hbar} = \sqrt{\frac{\mu^2 \omega_o}{2 \epsilon_0 \hbar V}} u(\vec{r}),
\end{equation} | \begin{equation}
g_o(\vec{r}) = \frac{\mu E}{\hbar} = \sqrt{\frac{\mu^2 \omega_o}{2 \epsilon_0 \hbar V}} u(\vec{r}),
\end{equation} | \begin{equation}
g_o(\vec{r}) = \frac{\mu E}{\hbar} = \sqrt{\frac{\mu^2 \omega_o}{2 \epsilon_0 \hbar V}} u(\vec{r}),
\end{equation} |
2512.05096v1 | \gamma_{o}=\sqrt{\frac{3\pi\epsilon_{0}\hbar c^{3}}{n\omega^{3}\tau_{\mathrm{rad}}}}, | \begin{equation}
\gamma_o = \sqrt{\frac{3\pi \epsilon_0 \hbar c^3}{n \omega^3 \tau_{\rm rad}}},
\end{equation} | \begin{equation}
\gamma_{o} = \sqrt{\frac{3\pi \epsilon_0 \hbar c^3}{n \omega^3 \tau_{\text{rad}}}},
\end{equation} | \begin{equation}
\gamma_{o} = \sqrt{\frac{3\pi \epsilon_0 \hbar c^3}{n \omega^3 \tau_{\text{rad}}}},
\end{equation} |
2512.05096v1 | u({\vec{r}})=\sqrt{\frac{\epsilon({\vec{r}})\left|E({\vec{r}})\right|^{2}}{\operatorname*{max}\left[\epsilon({\vec{r}})\left|E({\vec{r}})\right|^{2}\right]}}, | \begin{equation}
u(\vec{r}) = \sqrt{\frac{\epsilon(\vec{r}) \left|E(\vec{r})\right|^2}{\max\left[\epsilon(\vec{r}) \left|E(\vec{r})\right|^2\right]}},
\end{equation} | \begin{equation}
u(\vec{r}) = \sqrt{\frac{\epsilon(\vec{r})|E(\vec{r})|^2}{\max\left[\epsilon(\vec{r})|E(\vec{r})|^2\right]}}
\end{equation} | \begin{equation}
u(\vec{r}) = \sqrt{\frac{\epsilon(\vec{r})|E(\vec{r})|^2}{\max\left[\epsilon(\vec{r})|E(\vec{r})|^2\right]}}
\end{equation} |
2512.05096v1 | R(\omega)=\left|r(\omega)\right|^{2}=\left|1-\frac{2\kappa_{c}}{i\left(\omega-\omega_{c}\right)+\kappa+g^{2}/\left(i\left(\omega-\omega_{a}\right)+\gamma\right)}\right|^{2}, | \begin{equation}
R(\omega)=\left|r(\omega)\right|^2=\left|1-\frac{2\kappa_c}{i\left(\omega-\omega_c\right)+\kappa+g^2/\left(i\left(\omega-\omega_a\right)+\gamma\right)}\right|^2,
\end{equation} | \begin{equation}
R(\omega) = \left|r(\omega)\right|^2 = \left|1 - \frac{2\kappa_c}{i(\omega - \omega_c) + \kappa + g^2 / (i(\omega - \omega_a) + \gamma)}\right|^2,
\end{equation} | \begin{equation}\\label{refspec}
\\mathcal{R}(\\omega)=|r(\\omega)|^2=\\left|1-\\frac{2 \\kappa_c}{i(\\omega-\\omega_c)+\\kappa+\\dfrac{g^2}{i(\\omega-\\omega_a)+\\gamma}}\\right|^2,
\\end{equation} |
2512.05096v1 | r_{\pm}=\pm i\Delta\underline{{{\Lambda^{\prime}{}^{2}\!R_{c}^{2}+\Delta\!\Lambda^{2}\left(\!\!{\kappa_{c}^{2}-\kappa_{i}^{2}\right)-\gamma\kappa_{c}}}}}. | \begin{equation}
r_{\pm} = \pm i \Delta \underline{\Lambda^{\prime 2} R_c^2 + \Delta \Lambda^2 \left( \kappa_c^2 - \kappa_i^2 \right) - \gamma \kappa_c}.
\end{equation} | \begin{equation}
r_\pm = \pm i \sqrt{\Delta^2 \Lambda^2 \left( \kappa_c^2 - \kappa_i^2 \right) - \gamma \kappa_c}.
\end{equation} | \begin{equation}
r_{\\pm}=\\pm i \\Delta \\frac{\\sqrt{\\gamma^2 \\kappa_c^2+\\Delta^2\\left(\\kappa_c^2-\\kappa_i^2\\right)}-\\gamma \\kappa_c}{\\Delta^2\\left(\\kappa_c+\\kappa_i\\right)}.
\\end{equation} |
2512.05096v1 | r_{\mathrm{ON}}=\frac{\left({1-K i/{\kappa_{c}}}\right)-C}{\left({1+K i/{\kappa_{c}}}\right)+C},\;\;\;\;\;\;\;\;\;\;r_{\mathrm{OFF}}=\frac{{1-k i/{\kappa_{c}}}}{{1+{\kappa_{i}}/{\kappa_{c}}}}, | \begin{equation}
r_{\text{ON}} = \frac{\left(1 - K i / \kappa_c\right) - C}{\left(1 + K i / \kappa_c\right) + C}, \quad r_{\text{ long}} = \frac{1 - k i / \kappa_c}{1 + k i / \kappa_c},
\end{equation} | \begin{equation}
r_{\text{ON}} = \frac{\left(1-K i/\kappa_c\right)-C}{\left(1+K i/\kappa_c\right)+C}, \quad r_{\text{OFF}} = \frac{1-K i/\kappa_c}{1+K i/\kappa_c},
\end{equation} | $$r_{\\text{ON}}=\\frac{(1-\\kappa_i/\\kappa_c)-\\mathcal{C}}{(1+\\kappa_i/\\kappa_c)+\\mathcal{C}},\\quad r_{\\text{OFF}}=\\frac{1-\\kappa_i/\\kappa_c}{1+\\kappa_i/\\kappa_c},$$ |
2512.05096v1 | \int_{-\infty}^{\infty}r(\omega)|f(\omega)|^{2}d\omega=\int_{-\infty}^{\infty}r(\omega)\left(G_{0}\delta(\omega)-\pi\tilde{G}(0)\delta^{(2)}(\omega)+...\right)=G_{0}r(0)-\pi\tilde{G}(0)r^{\prime\prime}(0)+... | \begin{equation}
\int_{-\infty}^{\infty}r(\omega)|f(\omega)|^2d\omega=\int_{-\infty}^{\infty}r(\omega)\left(G_0\delta(\omega)-\pi\tilde{G}(0)\delta^{(2)}(\omega)+...\right)=G_0r(0)-\pi\tilde{G}(0)r''(0)+...
\end{equation} | \begin{equation}
\int_{-\infty}^{\infty} r(\omega) |f(\omega)|^2 d\omega = \int_{-\infty}^{\infty} r(\omega) \left(G_0 \delta(\omega) - \pi \tilde{G}(0) \delta^{(2)}(\omega) + \ldots \right) = G_0 r(0) - \pi \tilde{G}(0) r''(0) + \ldots
\end{equation} | \begin{equation}
\\int_{-\\infty}^{\\infty} r(\\omega)|f(\\omega)|^2 d\\omega = \\\n\\int_{-\\infty}^{\\infty} r(\\omega)\\left(G_0 \\delta(\\omega) - \\pi \\ddot{G}(0) \\delta^{(2)}(\\omega) + \\cdots\\right) = G_0 r(0) -\\pi \\ddot{G}(0) r''(0) + \\cdots
\\end{equation} |
2512.05096v1 | F_{\mathrm{signal}}\approx\left|G_{0}r(0)-\pi\dot{{G}}(0)r^{\prime\prime}(0)\right|^{2}. | \begin{equation}
F_{\text{ signal}} \approx \left|G_0 r(0) - \pi \ddot{G}(0) r''(0) \right|^2.
\end{equation} | \begin{equation}
F_{\rm signal} \approx \left|G_0 r(0) - \pi \dot{G}(0) r''(0)\right|^2.
\end{equation} | \begin{equation}
F_{\\text{signal}} \\approx \\left| G_0 r(0) - \\pi \\ddot{G}(0) r''(0) \\right|^2.
\\end{equation} |
2512.05096v1 | F_{\mathrm{signal}}=\left|\int_{-\infty}^{\infty}r(\omega)|f(\omega)|^{2}d\omega\right|^{2}, | \begin{equation}
F_{\text{ signal}} = \left|\int_{-\infty}^{\infty} r(\omega) |f(\omega)|^2 d\omega \right|^2,
\end{equation} | \begin{equation}
F_{\mathrm{signal}} = \left| \int_{-\infty}^{\infty} r(\omega) |f(\omega)|^2 d\omega \right|^2,
\end{equation} | \begin{equation}
F_{\\text{signal}} = \\left| \\int_{-\\infty}^{\\infty} r(\\omega)|f(\\omega)|^2 d\\omega \\right|^2.
\\end{equation} |
2512.05096v1 | \begin{array}{l}{{{\mathcal Z}\left(\left|0\right\rangle\left\langle0\right|\right\rangle=\left|0\right\rangle\left\langle0\right|,}}\\ {{{\mathcal E}\left(\left|0\right\rangle\left\langle1\right|\right)=\sqrt{1-p_{a}}\left(1-p_{d}\right)\left|0\right\rangle\left\langle0\right|,}}\\ {{{\mathcal E}\left(\left|1\right\ra... | \begin{align}
\mathcal{Z}(\lvert 0 \rangle \langle 0 \rvert) &= \lvert 0 \rangle \langle 0 \rvert, \\
\mathcal{E}(\lvert 0 \rangle \langle 1 \rvert) &= \sqrt{1-p_a}(1-p_d) \lvert 0 \rangle \langle 0 \rvert, \\
\mathcal{E}(\lvert 1 \rangle \langle 0 \rvert) &= p_a \lvert 0 \rangle \langle 0 \rvert + (1-p_a) ... | \begin{align}
\mathcal{Z}(\left|0\right\rangle\left\langle0\right|) &= \left|0\right\rangle\left\langle0\right|, \notag \\
\mathcal{E}(\left|0\right\rangle\left\langle1\right|) &= \sqrt{1-p_a}\left(1-p_d\right)\left|0\right\rangle\left\langle0\right|, \notag \\
\mathcal{E}(\left|1\right\rangle\left\langle0\... | \[\begin{aligned}
\mathcal{E}(\lvert 0\rangle\langle 0\rvert) &= \lvert 0\rangle\langle 0\rvert, \\
\mathcal{E}(\lvert 0\rangle\langle 1\rvert) &= \sqrt{1 - p_a(1 - p_d)} \\;\lvert 0\rangle\langle 1\rvert, \\
\mathcal{E}(\lvert 1\rangle\langle 0\rvert) &= \sqrt{1 - p_a(1 - p_d)} \\;\lvert 1\rangle\langle 0\rvert, \\
\m... |
2512.05096v1 | \mathcal{F}_{\mathrm{mw}}=\frac{1}{4}\left(1+\big(1-p_{a}\big)+2\sqrt{1-p_{a}}\big(1-p_{d}\big)\right). | \begin{equation}
\mathcal{F}_{\text{w}} = \frac{1}{4} \left(1 + \big(1 - p_a\big) + 2 \sqrt{1 - p_a} \big(1 - p_d\big)\right).
\end{equation} | \begin{equation}
\mathcal{F}_{\mathrm{mw}} = \frac{1}{4}\left(1 + (1-p_a) + 2\sqrt{1-p_a}(1-p_d)\right).
\end{equation} | \begin{equation}
\\mathcal{F}_{\\text{mw}} = \\frac{1}{4}\\left(1 + (1-p_a)+2\\sqrt{1-p_a}(1-p_d)\\right).
\\end{equation} |
2512.05096v1 | F_{\mathrm{mw}}=\frac{1}{4}\big(1+e^{-\tau/T_{1,\mathrm{t}}}+\Sigma e^{-\tau/2T_{1,\mathrm{t}}}e^{-\tau/T_{2,\mathrm{s}}}\big). | \begin{equation}
F_{\text{w}} = \frac{1}{4} \big(1 + e^{-\tau/T_{1,\text{t}}} + \Sigma e^{-\tau/2T_{1,\text{t}}} e^{-\tau/T_{2,\text{s}}}\big).
\end{equation} | \begin{equation}
F_{\rm mw} = \frac{1}{4}\big(1+e^{-\tau/T_{1,\rm t}}+\Sigma e^{-\tau/2T_{1,\rm t}}e^{-\tau/T_{2,\rm s}}\big).
\end{equation} | \begin{equation}
F_{\rm mw} = \frac{1}{4}\big(1+e^{-\tau/T_{1,\rm t}}+\Sigma e^{-\tau/2T_{1,\rm t}}e^{-\tau/T_{2,\rm s}}\big).
\end{equation} |
2512.05096v1 | \rho=|\Phi\rangle\left\langle\Phi\right|=\frac{1}{2}\left(|E\rangle\left\langle E\right|_{o}\left|1\right\rangle\left\langle1\right|_{m}+|E\right\rangle\left\langle L\right|_{o}\left|0\right|_{m}+\left|L\right\rangle\left\langle E\right|_{o}\left|0\right\rangle\left\langle0\right|_{m}+\left|L\right\rangle\left\langle L... | \begin{equation}
\rho = |\Phi\rangle \left\langle \Phi \right| = \frac{1}{2} \left( |E\rangle \left\langle E \right|_o \left|1\right\rangle \left\langle 1 \right|_m + |E\rangle \left\langle L \right|_o \left|0\right\rangle \left\langle 0 \right|_m + \left|L\right\rangle \left\langle E \right|_o \left|0\right\rangle... | \begin{equation}
\rho = |\Phi\rangle\langle\Phi| = \frac{1}{2}\left(|E\rangle\langle E|_o|1\rangle\langle 1|_m + |E\rangle\langle L|_o|0\rangle\langle 0|_m + |L\rangle\langle E|_o|0\rangle\langle 0|_m + |L\rangle\langle L|_o|0\rangle\langle 0|_m\right).
\end{equation} | \begin{equation}
\\rho = \\ket{\\Phi}\\bra{\\Phi}
= \\frac{1}{2}\\Big(
\\ket{E}\\bra{E}_o \\, \\ket{1}\\bra{1}_m
+ \\ket{E}\\bra{L}_o \\, \\ket{1}\\bra{0}_m
+ \\ket{L}\\bra{E}_o \\, \\ket{0}\\bra{1}_m
+ \\ket{L}\\bra{L}_o \\, \\ket{0}\\bra{0}_m
\\Big).
\\end{equation} |
2512.05096v1 | \hat{\cal H}^{\prime}\approx\hat{\cal H}_{H F}=\frac{1}{4}{\cal A}_{\parallel}\hat{\sigma}_{z}^{S}\hat{\sigma}_{z}^{I}+\frac{1}{4}{\cal A}_{\perp}(\hat{\sigma}_{x}^{S}\hat{\sigma}_{x}^{I}+\hat{\sigma}_{y}^{S}\hat{\sigma}_{y}^{I}), | \begin{equation}
\hat{H}^\prime \approx \hat{H}_{H F} = \frac{1}{4} \mathcal{A}_\parallel \hat{\sigma}_z^S \hat{\sigma}_z^I + \frac{1}{4} \mathcal{A}_\perp (\hat{\sigma}_x^S \hat{\sigma}_x^I + \hat{\sigma}_y^S \hat{\sigma}_y^I),
\end{equation} | \begin{equation}
\hat{\mathcal{H}}' \approx \hat{\mathcal{H}}_{HF} = \frac{1}{4} \mathcal{A}_\parallel \hat{\sigma}^S_z \hat{\sigma}^I_z + \frac{1}{4} \mathcal{A}_\perp (\hat{\sigma}^S_x \hat{\sigma}^I_x + \hat{\sigma}^S_y \hat{\sigma}^I_y),
\end{equation} | \begin{equation}
\\Hamilo = \\frac{1}{2}\\lambda \\Op{z}^L \\Op{z}^S - \\frac{1}{2} (\\alpha_{E_{gx}} \\Op{x}^L + \\alpha_{E_{gy}} \\Op{y}^L).
\\end{equation} |
2512.05096v1 | \hat{\cal H}_{0}=\frac{1}{\overline{{{\bf2}}}}{\lambda}\hat{\sigma}_{z}^{L}\hat{\sigma}_{z}^{S}-\frac{1}{\bf2}\big({\bf q}_{E}{\bf s}_{g x}\hat{\cal O}_{x}^{L}+{\bf q}_{E}_{g y}\hat{\sigma}_{y}^{L}\big). | \begin{equation}
\hat{H}_0 = \frac{1}{\overline{2}} \lambda \hat{\sigma}_z^L \hat{\sigma}_z^S - \frac{1}{\overline{2}} \big( \mathbf{q}_E \mathbf{s}_{gx} \hat{\sigma}_x^L + \mathbf{q}_E \mathbf{s}_{gy} \hat{\sigma}_y^L \big).
\end{equation} | \begin{equation}
\hat{\mathcal{H}}_0 = \frac{1}{2}\lambda \hat{\sigma}^L_z \hat{\sigma}^S_z - \frac{1}{2} \big( \mathbf{q}_{E x} \hat{\sigma}^L_x + \mathbf{q}_{E y} \hat{\sigma}^L_y \big).
\end{equation} | \begin{equation}
\\Hamilo = \\frac{1}{2}\\lambda \\Op{z}^L \\Op{z}^S - \\frac{1}{2} (\\alpha_{E_{gx}} \\Op{x}^L + \\alpha_{E_{gy}} \\Op{y}^L).
\\end{equation} |
2512.05096v1 | \big|\Psi_{1}\big>\,=\,c_{1}\left|+L,\left|\O_{2}\right>\Sigma\right>\right>-\,c_{2}e^{i\phi}\left|-L,\right|\left.\bar{S}\right>\bar{\psi}\right> | \begin{equation}
\big|\Psi_1\big\rangle = c_1 \big|+L,\big\rangle \big|\O_2\big\rangle \big\rangle - c_2 e^{i\phi} \big|-L,\big\rangle \big|\bar{S}\big\rangle \big\rangle
\end{equation} | \begin{equation}
\left|\Psi_1\right>=c_1\left|+L,\left|\bar{S}_2\right>\right> - c_2 e^{i\phi}\left|-L,\left|\bar{S}_2\right>\right>
\end{equation} | \begin{align}
\\ket{\\Psi_1} &= c_1 \\ket{+L, \\uparrow S} - c_2 e^{i\\phi} \\ket{-L, \\uparrow S} \\
\\ket{\\Psi_2} &= c_2 \\ket{+L, \\downarrow S} - c_1 e^{-i\\phi} \\ket{-L, \\downarrow S}
\\end{align} |
2512.05096v1 | \big|\Psi_{2}\big>=\,{\cal C}_{2}\,\big|+L,\llap/{\Psi}\mathrm{~}{\bf{\bar{\cal S}}}\big>-\,{\cal C}_{1}\,{\cal C}^{-\,i\phi}\,\big|-L,\forall\mathrm{~}{\bf{\bar{\cal S}}}\big> | \begin{equation}
\left|\Psi_2\right> = \mathcal{C}_2 \left|+L\right> - \mathcal{C}_1 \mathcal{C}^{-i\phi} \left|-L\right>, \quad \forall \left|\Psi_2\right> \in \mathcal{H}_{\mathrm{eff}}.
\end{equation} | \begin{equation}
\big|\Psi_2\big>=\,C_2\,\big|+L,\bm{\Psi}\big> -\,C_1\,C^{-i\phi}\,\big|-L,\bm{\Psi}\big>
\end{equation} | \\[\\ket{\\Psi_2} = c_2 \\ket{+L, \\downarrow S} - c_1 e^{-i\\phi} \\ket{-L, \\downarrow S}\\] |
2512.05096v1 | \left(\operatorname{E}{\mathfrak{T}}{\mathfrak{I}}\right) | noise | \left(\operatorname{E}{\mathfrak{T}}{\mathfrak{I}}\right) | noise |
2512.05096v1 | \hat{H}_{m_{J}=0}=\left(\begin{array}{c c}{{-A_{||}/4}}&{{\frac{A_{1}\alpha}{2\Delta}}}\\ {{\frac{A_{1}\alpha}{2\Delta}}}&{{-A_{||}/4}}\end{array}\right), | \begin{equation}
\hat{H}_{m_J=0}=\begin{pmatrix}
-A_{||}/4 & \frac{A_1\alpha}{2\Delta} \\
\frac{A_1\alpha}{2\Delta} & -A_{||}/4
\end{pmatrix},
\end{equation} | \begin{equation}
\hat{H}_{m_J=0}=\begin{pmatrix}
- A_{\parallel}/4 & \frac{A_1 \alpha}{2 \Delta} \\
\frac{A_1 \alpha}{2 \Delta} & - A_{\parallel}/4
\end{pmatrix},
\end{equation} | \begin{equation}
\hat{H}_{m_J=0}=\begin{pmatrix}
- A_{\parallel}/4 & \frac{A_1 \alpha}{2 \Delta} \\
\frac{A_1 \alpha}{2 \Delta} & - A_{\parallel}/4
\end{pmatrix},
\end{equation} |
2512.05096v1 | \left|{\cal E}_{m,J=0^{\prime\prime}}^{+}\right>=\frac{1}{\sqrt{2}}\left(\left|\Psi_{3},\right|\backslash\right>-\left|\Psi_{4},\left|\right|\right>\right). | \begin{equation}
\left|\mathcal{E}_{m,J=0''}^+\right>=\frac{1}{\sqrt{2}}\left(\left|\Psi_{3},\right|\backslash\right>-\left|\Psi_{4},\left|\right|\right).
\end{equation} | \begin{equation}
\left|E_{m,J=0^{\prime\prime}}^+\right>=\frac{1}{\sqrt{2}}\left(\left|\Psi_3,\right|\backslash\right>-\left|\Psi_4,\left|\right|\right>\right).
\end{equation} | \\[\\ket{E_{m_J=0'}^+} = \\frac{1}{\\sqrt{2}}(\\ket{\\Psi_3,\\downarrow} + \\ket{\\Psi_4,\\uparrow})\\] |
2512.05096v1 | \left|{\cal E}_{m n_{J}=0^{\prime}}^{+}\right>\,=\,\frac{1}{\sqrt{2}}\left(\left|\Psi_{3},\left|\right|\right>\,+\,\left|\Psi_{4},\,\right|\right>\right) | \begin{equation}
\left|\mathcal{E}_{mn_{J}=0'}^+\right\rangle = \frac{1}{\sqrt{2}}\left(\left|\Psi_3,\Psi_4\right\rangle + \left|\Psi_4,\Psi_3\right\rangle\right)
\end{equation} | \begin{equation}
\left|E_{m n_{J}=0^{\prime}}^{+}\right>=\frac{1}{\sqrt{2}}\left(\left|\Psi_{3},\left|\right|\right>+\left|\Psi_{4},\,\right|\right> \right)
\end{equation} | \\[\\ket{E_{m_J=0''}^+} = \\frac{1}{\\sqrt{2}}(\\ket{\\Psi_3,\\downarrow} - \\ket{\\Psi_4,\\uparrow})\\] |
2512.05096v1 | \big\vert\Psi_{3}\big\rangle=c_{2}e^{-i\phi}\left\vert+L,\uparrow\;S\right\rangle+c_{1}\left\vert-L,\uparrow\;S\right\rangle | \begin{equation}
\big|\Psi_3\big\rangle = c_2 e^{-i\phi} \big|+L,\uparrow\;S\big\rangle + c_1 \big|-L,\uparrow\;S\big\rangle
\end{equation} | \begin{equation}
\big| \Psi_3 \big\rangle = c_2 e^{-i\phi} \big| +L, \uparrow S \big\rangle + c_1 \big| -L, \uparrow S \big\rangle
\end{equation} | \\[\\ket{\\Psi_3} = c_2 e^{-i\\phi} \\ket{+L, \\uparrow S} + c_1 \\ket{-L, \\uparrow S}\\] |
2512.05096v1 | \big|\Psi_{4}\big\rangle=c_{1}e^{i\phi}\left|+L,\llap\rangle_{\cdot}\downarrow S\right\rangle+c_{2}\left|-L,\downarrow\mathrm{~}S\right\rangle | \begin{equation}
\big|\Psi_4\big\rangle = c_1 e^{i\phi} \left|+L,\downarrow S\right\rangle + c_2 \left|-L,\downarrow S\right\rangle
\end{equation} | \begin{equation}
\big|\Psi_4\big\rangle = c_1 e^{i\phi} \big|+L,\downarrow S\big\rangle + c_2 \big|-L,\downarrow S\big\rangle
\end{equation} | \\[\\ket{\\Psi_4} = c_1 e^{i\\phi} \\ket{+L, \\downarrow S} + c_2 \\ket{-L, \\downarrow S}\\] |
2512.05096v1 | P_{g1}=\rho_{11}(t=\infty)=\frac{\gamma\kappa(\gamma+\kappa)+4g^{2}(\gamma_{1}+\kappa)}{(\gamma+\kappa)\left(4g^{2}+\gamma\kappa\right)},\qquad P_{g_{2}}=\rho_{22}(t=\infty)=\frac{4g^{2}\gamma_{2}}{(\gamma+\kappa)\left(4g^{2}+\gamma\kappa\right)}. | \begin{equation}
P_{g1}=\rho_{11}(t=\infty)=\frac{\gamma\kappa(\gamma+\kappa)+4g^2(\gamma_1+\kappa)}{(\gamma+\kappa)\left(4g^2+\gamma\kappa\right)},\qquad P_{g_2}=\rho_{22}(t=\infty)=\frac{4g^2\gamma_2}{(\gamma+\kappa)\left(4g^2+\gamma\kappa\right)}.
\end{equation} | \begin{equation}
P_{g1} = \rho_{11}(t=\infty) = \frac{\gamma\kappa(\gamma+\kappa)+4g^2(\gamma_1+\kappa)}{(\gamma+\kappa)\left(4g^2+\gamma\kappa\right)}, \qquad P_{g2} = \rho_{22}(t=\infty) = \frac{4g^2\gamma_2}{(\gamma+\kappa)\left(4g^2+\gamma\kappa\right)}.
\end{equation} | \begin{equation}
P_{g1}= \\rho_{11} (t =\\infty) = \\frac{\\gamma \\kappa(\\gamma+\\kappa)+4 g^2(\\gamma_1+\\kappa)}{(\\gamma+\\kappa)\\left(4 g^2+\\gamma \\kappa\\right)}, \\quad \\quad P_{g_2}=\\rho_{22} (t =\\infty) = \\frac{4 g^2\\gamma_2}{(\\gamma+\\kappa)\\left(4 g^2+\\gamma \\kappa\\right)}.
\\end{equation} |
2512.05096v1 | \left({\overline{{\mathbf{I},}}}\bot\Theta\right) | noise | \left({\overline{{\mathbf{I},}}}\bot\Theta\right) | noise |
2512.05096v1 | \frac{d\hat{\sigma}_{g_{1}e}}{d t}=\left(-i\omega_{a}-\frac{\gamma}{2}\right)\hat{\sigma}_{g_{1}e}+i g\hat{\sigma}_{z}\hat{a} | \begin{equation}
\frac{d\hat{\sigma}_{g_1e}}{dt} = \left(-i\omega_a - \frac{\gamma}{2}\right)\hat{\sigma}_{g_1e} + i g \hat{\sigma}_z \hat{a}
\end{equation} | \begin{equation}
\frac{d \hat{\sigma}_{g_1e}}{dt} = \left(-i\omega_a - \frac{\gamma}{2}\right)\hat{\sigma}_{g_1e} + i g \hat{\sigma}_z \hat{a}
\end{equation} | \begin{equation}
\frac{d \hat{\sigma}_{g_1e}}{dt} = \left(-i\omega_a - \frac{\gamma}{2}\right)\hat{\sigma}_{g_1e} + i g \hat{\sigma}_z \hat{a}
\end{equation} |
2512.05096v1 | {\frac{d\hat{a}}{d t}}=\left(-i\omega_{c}-\kappa\right)\hat{a}-i g\hat{\sigma}_{g_{1}e}+\sqrt{2\kappa_{\mathrm{c}}}\hat{a}_{\mathrm{in}}\left(t\right) | \begin{equation}
\frac{d\hat{a}}{dt} = \left(-i\omega_c - \kappa\right)\hat{a} - i g \hat{\sigma}_{ge} + \sqrt{2\kappa_{\mathrm{c}}}\hat{a}_{\mathrm{in}}(t)
\end{equation} | \begin{equation}
\frac{d\hat{a}}{dt} = \left(-i\omega_c - \kappa\right)\hat{a} - i g \hat{\sigma}_{g_1e} + \sqrt{2\kappa_\text{c}}\hat{a}_\text{in}(t)
\end{equation} | \begin{equation}
\frac{d\hat{a}}{dt} = \left(-i\omega_c - \kappa\right)\hat{a} - i g \hat{\sigma}_{g_1e} + \sqrt{2\kappa_\text{c}}\hat{a}_\text{in}(t)
\end{equation} |
2512.05096v1 | R_{\mathrm{herald}}=0.5\cdot\frac{C^{2}}{\left(1+C\right)^{2}}\cdot\frac{\left(1-e^{-\Gamma\tau}\right)}{\tau+T_{\mathrm{reset}}}. | \begin{equation}
R_{\text{herd}} = 0.5 \cdot \frac{C^2}{(1+C)^2} \cdot \frac{1 - e^{-\Gamma \tau}}{\tau + T_{\text{ erase}}}.
\end{equation} | \begin{equation}
R_{\text{herald}} = 0.5 \cdot \frac{C^2}{(1+C)^2} \cdot \frac{(1 - e^{-\Gamma \tau})}{\tau + T_{\text{ reset}}}.
\label{eq:Rherald}
\end{equation} | \begin{equation}
R_{\\text{herald}} = 0.5 \\cdot \\frac{C^2}{(1+C)^2} \\cdot \\frac{(1-e^{-\\Gamma \\tau})}{\\tau + T_{\\text{reset}}}.
\\end{equation} |
2512.05101v1 | \begin{array}{c}{{\ell=\underline{{{s}}}_{0}^{Z}+\underline{{{s}}}_{1}^{Z}\left[1-h\left(\bar{\phi}_{1}^{Z}\right)\right]-f_{E C}n^{Z}h(E^{Z})}}\\ {{\overline{{{-\log}}_{2}\frac{2}{\varepsilon_{\mathrm{corr}}}-6\log_{2}\frac{21}{\varepsilon_{\mathrm{sec}}},}}\end{array} | \begin{align}
\ell = \underline{s}_0^Z + \underline{s}_1^Z \left[1 - h(\bar{\phi}_1^Z)\right] - f_{EC} n^Z h(E^Z)
\end{align} | \begin{align}
\ell = \underline{s}^Z_0 + \underline{s}^Z_1 \left[1 - h(\bar{\phi}^Z_1)\right] - f_{EC} n^Z h(E^Z) \notag \\
- \log_2 \frac{2}{\varepsilon_{\mathrm{corr}}} - 6 \log_2 \frac{21}{\varepsilon_{\mathrm{sec}}}, \notag
\end{align} | \begin{equation}
\\delta_{n>1}^{(1)} > \\delta_{n>1}^{(2)} > \\delta_{n>1}^{(0)}.
\\end{equation} |
2512.05101v1 | \langle{n_{1}}\rangle\ >\ \langle{n_{0}}\rangle\,+\,\langle{n_{2}}\rangle\quad\mathrm{and}\quad\ \langle{n_{2}}\rangle\,>\,\langle{n_{0}}\rangle\,\ge\,0, | \begin{equation}
\langle n_1 \rangle > \langle n_0 \rangle + \langle n_2 \rangle \quad \text{and} \quad \langle n_2 \rangle > \langle n_0 \rangle \ge 0,
\end{equation} | \begin{equation}
\langle n_1 \rangle > \langle n_0 \rangle + \langle n_2 \rangle \quad \text{and} \quad \langle n_2 \rangle > \langle n_0 \rangle \geq 0,
\end{equation} | \begin{equation} \label{eq:mean-requirement}
\mean{n_1} > \mean{n_0} + \mean{n_2}
\quad\mathrm{and}\quad
\mean{n_2}>\mean{n_0} \ge 0,
\end{equation} |
2512.05101v1 | \bar{\partial}_{n>1}^{(1)}\,\ >\,\bar{\partial}_{n>1}^{(2)}\,>\,\bar{\partial}_{n>1}^{(0)}, | \begin{equation}
\bar{\partial}_{n>1}^{(1)} \, > \, \bar{\partial}_{n>1}^{(2)} \, > \, \bar{\partial}_{n>1}^{(0)},
\end{equation} | \begin{equation}
\bar{\partial}^{(1)}_{n>1} > \bar{\partial}^{(2)}_{n>1} > \bar{\partial}^{(0)}_{n>1},
\end{equation} | \begin{equation}
\\frac{d\\rho}{dt} = -\\frac{i}{\\hbar}[H, \\rho] + \\sum_n \\frac{1}{2}(2C_n\\rho C_n^{\\dagger} - \\rho C_n^{\\dagger}C_n - C_n^{\\dagger}C_n\\rho) = \\mathcal{L}\\rho.
\\end{equation} |
2512.05101v1 | \frac{d\rho}{d t}=-\frac{i}{\hbar}[H,\rho]+\sum_{n}\frac{1}{2}[2C_{n}\rho C_{n}^{\dagger}-\rho C_{n}^{\dagger}C_{n}-C_{n}^{\dagger}C_{n}\rho]=Z\rho, | \begin{equation}
\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho] + \sum_n \frac{1}{2}[2C_n\rho C_n^\dagger - \rho C_n^\dagger C_n - C_n^\dagger C_n\rho] = Z\rho,
\end{equation} | \begin{equation}
\frac{d \rho}{dt} = -\frac{i}{\hbar}[H, \rho] + \sum_n \frac{1}{2}[2C_n\rho C_n^\dagger - \rho C_n^\dagger C_n - C_n^\dagger C_n\rho] = Z\rho,
\end{equation} | \begin{multline}
\\frac{p_{0|\\delta(b)} n_{X,\\delta(a)}}{p_{\\delta(a)}} - \\frac{p_{0|\\delta(a)} n_{X,\\delta(b)}}{p_{\\delta(b)}} = \\\n\\frac{n_{X,1}}{p_1}(p_{0|\\delta(b)}p_{1|\\delta(a)} - p_{0|\\delta(a)}p_{1|\\delta(b)}) + \\\n\\sum_{n=2}^{\\infty}\\frac{n_{X,n}}{p_n}(p_{0|\\delta(b)}p_{n|\\delta(a)} - p_{0|\... |
2512.05101v1 | \begin{array}{c}{{n_{X,1}\geq\frac{p_{1}}{p_{0}|\delta(b)p_{1}|\delta(a)-p_{0}|\delta(a)}\sum_{n=2}\frac{p_{0}|\delta(b)n_{X,\delta(a)}}{p_{\delta(b)}}}\\ {{-\frac{p_{0}|\delta(a)n_{X,\delta(b)}}{p_{\delta(b)}}-p_{0}|\delta(a)}}\end{array} | \begin{equation}
\begin{array}{c}
n_{X,1} \geq \frac{p_1}{p_0|\delta(b)p_1|\delta(a) - p_0|\delta(a)|} \sum_{n=2} \frac{p_0|\delta(b)n_{X,\delta(a)}}{p_{\delta(b)}} - \frac{p_0|\delta(a)n_{X,\delta(b)}|}{p_{\delta(b)}} - p_0|\delta(a)|
\end{array}
\end{equation} | \begin{align}
n_{X,1} \geq \frac{p_1}{p_0|\delta(b)p_1| - |\delta(a)p_0|} \sum_{n=2} \frac{p_0|\delta(b)n_{X,\delta(a)}}{p_{\delta(b)}} - \frac{p_0|\delta(a)n_{X,\delta(b)}}{p_{\delta(b)}} - p_0|\delta(a) \label{eq:lower_bound}
\end{align} | \begin{multline}
n_{X,1} \\ge \\frac{\\underline{p}_1}{\\overline{p}_{1|\\delta(c)} - \\underline{p}_{1|\\delta(a)}} \\Bigg( \\frac{n_{X,\\delta(c)}}{p_{\\delta(c)}} - \\frac{n_{X,\\delta(a)}}{p_{\\delta(a)}} \\\n - \\overline{p}_{0|\\delta(c)} \\frac{\\overline{n}_{X,0}}{\\underline{p}_0} + \\underline{p}_{0|\\delta(... |
2512.05101v1 | n_{X,1}\ge\frac{p_{1}}{\overline{{p}}_{1|\delta(c)}-\frac{p}{-1|\delta(a)}}\left(\frac{n_{X,\delta(c)}}{p_{\delta(c)}}-\frac{n_{X,\delta(a)}}{p_{\delta(a)}}\right), | \begin{equation}
n_{X,1} \ge \frac{p_1}{\overline{p}_{1|\delta(c)} - \frac{p}{-1|\delta(a)}} \left( \frac{n_{X,\delta(c)}}{p_{\delta(c)}} - \frac{n_{X,\delta(a)}}{p_{\delta(a)}} \right),
\end{equation} | \begin{equation}
n_{X,1} \geq \frac{p_1}{\overline{p}_{1|\delta(c)} - \frac{p_1}{\delta(a)}} \left( \frac{n_{X,\delta(c)}}{p_{\delta(c)}} - \frac{n_{X,\delta(a)}}{p_{\delta(a)}} \right),
\end{equation} | \begin{equation}
\\overline{n}_{X,0} = n_{X,\\delta(0)} \\left(1 + \\frac{p_{\\delta(a)}}{p_{\\delta(0)}} \\overline{p}_{0|\\delta(a)} + \\frac{p_{\\delta(c)}}{p_{\\delta(0)}} \\overline{p}_{0|\\delta(c)} \\right).
\\end{equation} |
2512.05101v1 | \begin{array}{l}{{\begin{array}{c}{{p_{0}|\delta(b)^{n}X,\delta(a)}}\\ {{p_{3}(a)}}\end{array}-\frac{p_{0}|\delta(a)^{n}X,\delta(b)}{p_{\delta(b)}}=\qquad\qquad\qquad\qquad}}\\ {{\qquad\qquad\sum_{n=2}\frac{n_{X,n}}{p_{n}}\Biggl(p_{0}|\delta(b)p_{1}|\delta(a)-p_{0}|\delta(a)p_{n}|\delta(b)\Biggr),}}\end{array} | \begin{align}
\begin{split}
&\begin{split}
p_0|\delta(b)^nX,\delta(a) - p_3(a) &= \frac{p_0|\delta(a)^nX,\delta(b)}{p_{\delta(b)}} = \sum_{n=2}\frac{n_X,n}{p_n}\Bigl(p_0|\delta(b)p_1|\delta(a) - p_0|\delta(a)p_n|\delta(b)\Bigr),
\end{split}
\end{split}
\end{split}
\end{align} | \begin{align}
\begin{array}{c}
p_0|\delta(b)^{n}X,\delta(a)\\
p_3(a)
\end{array}
-\frac{p_0|\delta(a)^{n}X,\delta(b)}{p_{\delta(b)}}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\... | \begin{multline}
n_{X,1} \\ge \\frac{p_1}{p_{0|\\delta(b)} p_{1|\\delta(a)} - p_{0|\\delta(a)} p_{1|\\delta(b)}} \\Bigg( \\frac{p_{0|\\delta(b)} n_{X,\\delta(a)}}{p_{\\delta(a)}} - \\frac{p_{0|\\delta(a)} n_{X,\\delta(b)}}{p_{\\delta(b)}} \\\n - p_{0|\\delta(b)} \\sum_{n=2}^{\\infty} \\frac{p_{n|\\delta(c)} n_{X,n}}{p... |
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