{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt 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","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"d6962f5a-0445-5e25-ae30-d0ad21b18d14","task_key":"train--Challenge~5f48~5fmain","task_revision_id":"2","upstream_id":"Challenge_48_main","short_description":"Let $Z(n, \\eta)$ denote a replica partition function defined for positive…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the expression of the analytic continuation $F(\\\\eta)$ in Sympy format\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    F_eta: float\\n        Analytic continuation $F(\\\\eta)$ at $\\\\eta = \\\\frac{10}{3} \\\\pi$, accurate to at least 8 decimal places.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    F_eta = ...\\n    # ---------------------------------------------------------------\\n\\n    return F_eta\",\"problem_description\":\"# Problem setup:\\nLet $Z(n, \\\\eta)$ denote a replica partition function defined for positive integers $n$ by\\n\\\\begin{equation}\\nZ(n, \\\\eta) = \\\\sum_{\\\\vec{x} \\\\in \\\\mathbb{Z}^{n-1}} \\\\exp\\\\left( -\\\\eta \\\\pi\\\\, \\\\vec{x}^\\\\top K \\\\vec{x} \\\\right),\\n\\\\end{equation}\\nwhere $\\\\eta > 0$ is a real parameter, and the sum runs over all integer vectors $\\\\vec{x} \\\\in \\\\mathbb{Z}^{n-1}$. The kernel $K$ is an $(n-1) \\\\times (n-1)$ matrix with components\\n\\\\begin{equation}\\nK_{ij} = \\\\left(1 - \\\\frac{1}{n}\\\\right)\\\\delta_{ij} - \\\\frac{1}{n}(1 - \\\\delta_{ij}).\\n\\\\end{equation}\\nEquivalently, $K = I_{n-1} - \\\\frac{1}{n} \\\\mathbf{1}_{n-1}$, where $\\\\mathbf{1}_{n-1}$ denotes the matrix of all ones.\\n\\n# Main problem:\\n\\nEvaluate the analytic continuation\\n\\\\begin{equation}\\nF( \\\\eta ) =  \\\\left. \\\\frac{\\\\partial}{\\\\partial n} Z(n, \\\\eta) \\\\right|_{n=1} - \\\\left( \\\\frac{1}{2} - \\\\frac{1}{2} \\\\ln \\\\eta \\\\right)\\n\\\\end{equation}\\nat $\\\\eta = \\\\frac{10}{3} \\\\pi$, accurate to at least eight digits after the decimal point.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}