\begin{algorithm}[H]
\caption{Expansion ranges}
  \label{alg:expansionrange}\begin{algorithmic}[1]
\State Input:
\begin{enumerate}
    \item[1] Difference equation system $\Sigma$ with algebraic or analytic transcendental functions.
    \item[2] For each parameter $\mu_i$ appearing as an argument of a transcendental function, user-specified acceptable domains for each $\mu_i$, denoted $R_{\mu_i}$.
    \item[3] Observed data $\bar{y}(t_0),\bar{y}(t_1),\ldots,\bar{y}(t_r)$
\end{enumerate}
\State Output: Domains $R_{G_{j}}$ over which each transcendental function $G_j$ in $\Sigma$ is expanded.
\Procedure{Expansion range}{} By substitution of maximums and minimums (modulus) of \phantom{-----} $R_{\mu_i}$ into arguments of $G_j$, as relevant, determine radii of discs $R_{L_t}$. 
\State Call the argument of $G_j$ as $\tilde{G}_j=:\tau_j$.
\For{$j=1\to r$, $t$ fixed}
\State $\displaystyle R_{G_{j,t}}:=\left\{\min_{R
_{\mu_i^*,i=1,\ldots,s} }|\tilde{G}_j|\le |\tau_j| \le\max_{R
_{\mu_i^*,i=1,\ldots,s} }|\tilde{G}_j|\right\}$
\EndFor 
\EndProcedure
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
[H]
\caption{Expansion ranges}
\begin{algorithmic}
[1]
\State Input:
\begin{enumerate}
\item[1] Difference equation system $\Sigma$ with algebraic or analytic transcendental functions.
\item[2] For each parameter $\mu_i$ appearing as an argument of a transcendental function, user-specified acceptable domains for each $\mu_i$, denoted $R_{\mu_i}$.
\item[3] Observed data $\bar{y}(t_0),\bar{y}(t_1),\ldots,\bar{y}(t_r)$
\end{enumerate}
\State Output: Domains $R_{G_{j}}$ over which each transcendental function $G_j$ in $\Sigma$ is expanded.
\Procedure{Expansion range}{} By substitution of maximums and minimums (modulus) of \phantom{-----} $R_{\mu_i}$ into arguments of $G_j$, as relevant, determine radii of discs $R_{L_t}$.
\State Call the argument of $G_j$ as $\tilde{G}_j=:\tau_j$.
\For{$j=1\to r$, $t$ fixed}
\State $\displaystyle R_{G_{j,t}}:=\left\{\min_{R
_{\mu_i^*,i=1,\ldots,s} }|\tilde{G}_j|\le |\tau_j| \le\max_{R
_{\mu_i^*,i=1,\ldots,s} }|\tilde{G}_j|\right\}$
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
{
    "title": "symbolic-numeric algorithm for parameter estimation in discrete-time   models with $\\exp$",
    "id": "2401.16220",
    "abstract": "determining unknown parameter values in dynamic models is crucial for accurate analysis of the dynamics across the different scientific disciplines. discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. in this paper, we propose a robust symbolic-numeric approach for parameter estimation in discrete-time models that involve non-algebraic functions such as exp. we illustrate the performance (precision) of our approach by applying our approach to the flour beetle (lpa) model, an archetypal discrete-time model in biology. unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values given time-series data for the measured variables.",
    "categories": "q-bio.qm cs.sc cs.sy eess.sy math.ac math.ds",
    "doi": "",
    "created": "2024-01-29",
    "updated": "",
    "authors": [
        "yosef berman",
        "joshua forrest",
        "matthew grote",
        "alexey ovchinnikov",
        "sonia rueda"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.16220"
}
{
    "ToxicityFilter": {
        "text_to_toxicity_attributes": {
            "arxiv-submission-Jan-29-2024.bbl": {
                "toxicity_score": 0.011874928,
                "severe_toxicity_score": 0.0011348724,
                "identity_attack_score": 0.003866276,
                "insult_score": 0.0075508766,
                "profanity_score": 0.014532468,
                "threat_score": 0.007521313,
                "sexually_explicit_score": 0.010969753,
                "flirtation_score": 0.2762957
            },
            "arxiv-submission-Jan-29-2024.tex": {
                "toxicity_score": 0.030270588,
                "severe_toxicity_score": 0.002861023,
                "identity_attack_score": 0.0055866763,
                "insult_score": 0.0116169015,
                "profanity_score": 0.03239325,
                "threat_score": 0.0097738765,
                "sexually_explicit_score": 0.024888365,
                "flirtation_score": 0.2879354
            }
        },
        "reason": null
    }
}
{
    "num_done": {
        "figure": 0,
        "algorithm": 3,
        "plot": 0
    }
}
{
    "NonTrivialRenderingFilter": {
        "white_pixels_ratio": 91.2402906931312,
        "hash": "1fbe800f201fdf3f",
        "most_frequent_color_ratio": 91.2402906931312
    }
}