{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex"},"task_id":"112c22b8-d4b4-5dda-aa61-40533999670d","task_key":"train--Challenge~5f2~5fmain","task_revision_id":"1","upstream_id":"Challenge_2_main","short_description":"Consider a population of genetically identical bacterial cells in balanced…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nlambda_plus, lambda_minus = sp.symbols('lambda_plus lambda_minus')\\nk_plus, k_minus = sp.symbols('k_plus k_minus')\\nalpha = sp.symbols('alpha')\\nvbar_b = sp.symbols('vbar_b')\\nbeta = sp.symbols('beta')\\nsigma2 = sp.symbols('sigma2')\\n\\ndef answer(lambda_plus, lambda_minus, k_plus, k_minus, alpha, vbar_b, beta, sigma2):\\n    r\\\"\\\"\\\"\\n    Return the expression of $\\\\Lambda$ in Sympy format, and the answer to the multiple choice question.\\n\\n    Inputs\\n    ----------\\n    lambda_plus : sympy.Symbol, growth-rate state $\\\\lambda^{+}$\\n    lambda_minus : sympy.Symbol, growth-rate state $\\\\lambda^{-}$\\n    k_plus, k_minus, alpha: sympy.Symbol, parameters of the gamma-distribution\\n    vbar_b: sympy.Symbol, average birth size, $\\\\bar v_b$\\n    beta: sympy.Symbol, parameter determining the degree of cell-size regulation, $0<\\\\beta\\\\leq 1$\\n    sigma2: sympy.Symbol, variance of the division noise, $\\\\sigma^2$\\n\\n    Outputs\\n    ----------\\n    Lambda : sympy.Expr, asymptotic population growth rate $\\\\Lambda$ to first order in $\\\\sigma^2/\\\\bar v_b^2$.\\n    answer_beta, answer_sigma2 : str, answers to the following multiple choice question.\\n        How $\\\\beta$ and $\\\\sigma^2$ affect the population growth rate?\\n          A. Increase B. Decrease C. Not affected D. Change nonmonotonically\\n        answer_beta: the answer for $\\\\beta$, one of {'A', 'B', 'C', 'D'}\\n        answer_sigma2: the answer for $\\\\sigma^2$, one of {'A', 'B', 'C', 'D'}\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    Lambda = ...  # a SymPy expression of inputs\\n    answer_beta = ...  # one of {'A', 'B', 'C', 'D'}\\n    answer_sigma2 = ...  # one of {'A', 'B', 'C', 'D'}\\n    # ---------------------------------------------------------------\\n\\n    return Lambda, answer_beta, answer_sigma2\",\"problem_description\":\"# Problem setup:\\nConsider a population of genetically identical bacterial cells in balanced growth. Each cell starts with some initial size $v_b$ and grows according to the equation\\n\\\\begin{equation}\\n    \\\\frac{dv}{dt} = \\\\lambda_t v(t),\\n\\\\end{equation}\\nwhere the growth rate $\\\\lambda_t$ is a two-state stochastic process that jumps between values $\\\\lambda^+$ and $\\\\lambda^-$ and has the gamma-distributed waiting times with densities\\n\\\\begin{equation}\\n    f_\\\\pm(t) = \\\\frac{k_\\\\pm^{\\\\alpha}\\\\, t^{\\\\alpha-1} e^{-k_\\\\pm t}}{\\\\Gamma(\\\\alpha)}.\\n\\\\end{equation}\\nEach cell divides symmetrically when it reaches a final division size given by\\n\\\\begin{equation}\\n    v_d = 2v_b^{1-\\\\beta}\\\\bar v_b^\\\\beta + \\\\xi.\\n\\\\end{equation}\\nHere, $v_b$ is the birth size of the cell, $\\\\bar v_b>0$ is a constant representing average birth size, $0<\\\\beta\\\\leq 1$ is a parameter determining the degree of cell-size regulation, and the division noise $\\\\xi>0$ is a narrowly distributed random variable with mean zero and variance $\\\\sigma^2$. You can assume this noise is Gaussian distributed and sufficiently narrow to ignore the probability that $v_d$ is ever smaller than $v_b$.\\n\\nA population of such cells grows asymptotically exponentially with growth rate $\\\\Lambda$, i.e., $N(t) \\\\propto e^{\\\\Lambda t}$ for large $t$.\\n\\n# Main problem:\\n\\nFind the asymptotic population growth rate $\\\\Lambda$ in terms of the model parameters $\\\\lambda^+$, $\\\\lambda^-$, $k_+$, $k_-$, $\\\\alpha$, $\\\\bar v_b$, $\\\\beta$, and $\\\\sigma^2$ for small $\\\\sigma^2/\\\\bar v_b^2$. Give your answer to first order in $\\\\sigma^2/\\\\bar v_b^2$. Explain how $\\\\beta$ and $\\\\sigma^2$ affect the population growth rate.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}