{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"scicode","formal_name":"SciCode","introduction":"SciCode evaluates the ability to solve scientific research problems through code. Problems are decomposed into subproblems; this dev import preserves the relationships between 15 parent problems and 50 subproblems.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/SciCode1/SciCode","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"8c3efb97-155a-58b9-a06e-4bfb5fb70743","task_key":"dev--49c68bb1-a088-53d9-a366-0885b5d1b63b--70~2e7","task_revision_id":"2","upstream_id":"70.7","short_description":"Compute the expansion coefficients $u_k$ of the evolution operator of the three…","config":"","split":"dev","body":"{\"step_background\":\"Background\\n\\nThe evolution operator is $\\\\mathbb{U}_{3}(L)=e^{-i H_{3} L}=$ $e^{-i h_{0} \\\\mathbb{1} L} e^{-i h_{k} \\\\lambda^{k} L}$. If we discard the global phase, we are left with $\\\\mathbb{U}_{3}(L)=e^{-i h_{k} \\\\lambda^{k} L}$.\\n\\nWe wish to expand $\\\\mathbb{U}_{3}$ using an identity for the Gell-Mann matrices:\\n\\n$$\\n\\\\begin{equation*}\\n\\\\mathbb{U}_{3}(L)=u_{0} \\\\mathbb{1}+i u_{k} \\\\lambda^{k}\\n\\\\end{equation*}\\n$$\\n\\nwhere the complex coefficients $u_{0}$ and $u_{k}$ are functions of $L$ and the $h_{k}$.\\n\\nThe coefficients in the above equation can be written as $u_{0}=\\\\frac{1}{3} \\\\operatorname{Tr} \\\\mathbb{U}_{3}$ and $u_{k}=-\\\\frac{i}{2} \\\\operatorname{Tr}\\\\left(\\\\lambda^{k} \\\\mathbb{U}_{3}\\\\right)$\\n\\nAn application of Sylvester's formula to $3 \\\\times 3$ matrices allows us to express the coefficients in terms of the $\\\\mathrm{SU}(3)$ invariants\\n\\n$$\\n\\\\begin{aligned}\\n& L^{2}|h|^{2} \\\\equiv L^{2} h_{k} h^{k} \\\\\\\\\\n& -L^{3}\\\\langle h\\\\rangle \\\\equiv-L^{3} d_{i j k} h^{i} h^{j} h^{k}\\n\\\\end{aligned}\\n$$\\n\\nNext, we solve the characteristic equation of $-h_{k} \\\\lambda^{k} L$, i.e., \\n$$\\\\phi^{3}-\\\\left(L^{2}|h|^{2}\\\\right) \\\\phi-\\\\frac{2}{3}\\\\left(-L^{3}\\\\langle h\\\\rangle\\\\right)=0$$. \\n\\nThe equation follows from the Cayley-Hamilton theorem, written conveniently in terms of invariants $\\\\langle h\\\\rangle$ and $|h|^{2}$. Its three latent roots, or eigenvalues, are $\\\\phi_{m} \\\\equiv \\\\psi_{m} L(m=1,2,3)$, with\\n\\n$$\\n\\\\begin{equation*}\\n\\\\psi_{m} \\\\equiv \\\\frac{2|h|}{\\\\sqrt{3}} \\\\cos \\\\left[\\\\frac{1}{3}(\\\\chi+2 \\\\pi m)\\\\right]\\n\\\\end{equation*}\\n$$\\n\\nThe expansion coefficient $u_k$ can be expressed as:\\n$$\\n\\\\begin{align*}\\n& u_{0}=\\\\frac{1}{3} \\\\sum_{m=1}^{3} e^{i L \\\\psi_{m}}\\\\\\\\\\n& u_{k}=\\\\sum_{m=1}^{3} e^{i L \\\\psi_{m}} \\\\frac{\\\\psi_{m} h_{k}-(h * h)_{k}}{3 \\\\psi_{m}^{2}-|h|^{2}}\\n\\\\end{align*}\\n$$\\n\\nwhere $(h * h)_{i} \\\\equiv d_{i j k} h^{j} h^{k}$.\",\"step_description_prompt\":\"Compute the expansion coefficients $u_k$ of the evolution operator of the three neutrino oscillation problem $\\\\mathbb{U}_{3}(L)=e^{-i h_{k} \\\\lambda^{k} L}$, if we expand it using the Gell-Mann matrices in the following way: $\\\\mathbb{U}_{3}(L)=u_{0} \\\\mathbb{1}+i u_{k} \\\\lambda^{k}$. Consider all coefficients from $k=0$ to $k=8$. $L$ is the baseline (i.e., the distance the neutrino has travelled in unit of eV).\"}","display_format":"scicode-step","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/SciCode1/SciCode","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}