| { |
| "version": 1, |
| "prompts": [ |
| { |
| "task_id": "set_missing_element-9223", |
| "task_prompt": "Set_A: ['vu', 'ui', 'ags', 'xh', 'aff', 'mn', 'wn', 'nh', 'wd', 'acx', 'qg', 'lm', 'vj', 'aag', 'ob', 'adz', 'aef', 'mt', 'ns', 'ahy', 'yl', 'le', 'qj', 'zb', 'abp', 'zu', 'ain', 'xm', 'aej', 'ahw', 'yd', 'acz', 'aib', 'afi', 'or', 'uz', 'we', 'sy', 'zt', 'ado', 'wa', 'op', 'xx', 'wo', 'ny', 'mh', 'acg', 'agt', 'nz', 'ql', 'vw', 'vf', 'aik', 'aab', 'pi', 'yu', 'km', 'yv', 'aig', 'uk', 'abj', 'abn', 'nl', 'sr', 'pk', 'wj', 'aai', 'adi', 'rf', 'yi', 'aiz', 'vm', 'tk', 'qo', 'afn', 'ahs', 'sx', 'ry', 'afv', 'agl', 've', 'xg', 'adn', 'xw', 'kt', 'um', 'pv', 'qc', 'lv', 'lj', 'adh', 'tp', 'qh', 'tx', 'lx', 'aeg', 'aey', 'afe', 'ko', 'na', 'vx', 'agc', 'kk', 'xv', 'kz', 'me', 'afh', 'qy', 'lg', 'ri', 'tn', 'yo', 'pa', 'qs', 'aak', 'agn', 'aes', 'ru', 'uw', 'acs', 'qa', 'wh', 'tj', 'sq', 'zq', 'wy', 'aev', 'vk', 'pf', 'zp', 'ahq', 'adu', 'adx', 'abi', 'pt', 'agz', 'ky', 'rr', 'oa', 'aby', 'sa', 'kw', 'abu', 'rt', 'rn', 'sp', 'acj', 'aeo', 'adv', 'aho', 'ks', 'xy', 'np', 'ahg', 'aaf', 'sd', 'ws', 'adk', 'lq', 'yb', 'ts', 'zj', 'xo', 'afb', 'tt', 'ahe', 'adw', 'afl', 'ahn', 'yp', 'wb', 'nn', 'adl', 'ady', 'ta', 'vd', 'ahp', 'ln', 'pm', 'agw', 'se', 'sw', 'nf', 'mq', 'on', 'zw', 'to', 'qi', 'acf', 'aao', 'adr', 'zy', 'aie', 'nr', 'zz', 'ya', 'agv', 'qz', 'va', 'yw', 'mr', 'xr', 'age', 'afx', 'ads', 'vy', 'zk', 'ais', 'zg', 'acu', 'sz', 'zc', 'aii', 'aer', 'zv', 'po', 'acq', 'ti', 'yy', 'ait', 'afd', 'afm', 'yt', 'xs', 'pn', 'st', 'abz', 'lw', 'nw', 'ahm', 'rz', 'rs', 'mv', 'no', 'agm', 'aic', 'wu', 'adt', 'aco', 'xi', 'zr', 'aid', 'oi', 'qn', 'tr', 'aay', 'aas', 'abl', 'ahh', 'ot', 'vs', 'si', 'wz', 'lr', 'ahd', 'lp', 'xb', 'ps', 'uv', 'aip', 'kq', 'agk', 'abm', 'aam', 'mm', 'os', 'acr', 'agx', 'aed', 'acd', 'sl', 'pb', 'vt', 'agb', 'aio', 'aee', 'uy', 'td', 'aba', 'pu', 'tb', 'afk', 'ol', 'aht', 'aax', 'tg', 'rj', 'lf', 'oz', 'uo', 'xt', 'pq', 'vr', 'adm', 'zi', 'tz', 'aat', 'abf', 'vb', 'nv', 'pg', 'abh', 'abc', 'mj', 'ul', 'agg', 'ut', 'afs', 'pl', 'agy', 'ade', 'zh', 'wm', 'sc', 'vv', 'vp', 'qp', 'lt', 'yh', 'zo', 'rv', 'nk', 'afc', 'adq', 'kn', 'py', 'tf', 'xf', 'ahv', 'll', 'aet', 'sh', 'afz', 'adf', 'agr', 'aar', 'lc', 'oe', 'aiw', 'zl', 'ra', 'sb', 'ahk', 'afp', 'mx', 'wf', 'yk', 'oc', 'agf', 'aaj', 'aiu', 'ml', 'om', 'sj', 'ye', 'vc', 'nd', 'un', 'ada', 'agq', 'rx', 'aiq', 'vo', 'qt', 'ld', 'oj', 'aca', 'zx', 'my', 'ahz', 'us', 'aei', 'ach', 'aaz', 'qv', 'tm', 'xz', 'aau', 'aen', 'pc', 'ub', 'adg', 'xp', 'tl', 'aem', 'vg', 'aaa', 'kr', 'od', 'ox', 'ro', 'oh', 'abv', 'uq', 'ph', 'nq', 'aif', 'agj', 'yq', 'tq', 'pz', 'aia', 'wq', 'vq', 'agh', 'of', 'afr', 'zm', 'up', 'ail', 'nb', 'kp', 'aae', 'agd', 'rp', 'ael', 'ss', 'qr', 'abr', 'oy', 'og', 'rc', 'ack', 'sn', 'abq', 'xl', 'yn', 'oq', 'tw', 'ua', 'afa', 'aeb', 'ahi', 'xq', 'ku', 'aga', 'rb', 'qw', 'ys', 'rk', 'mu', 'ug', 'ux', 'afg', 'acl', 'pw', 'mi', 'uc', 'abt', 'nc', 'qe', 'sv', 'zd', 'xe', 'aea', 'uj', 'ace', 'yg', 'ok', 'lb', 'pj', 'pd', 'agi', 'za', 'zf', 'aec', 'aiv', 'so', 'aan', 'abg', 'mk', 'vl', 'vz', 'abb', 'la', 'air', 'lo', 'ly', 'ww', 'kl', 'abk', 'vi', 'nu', 'rw', 'uu', 'acn', 'qk', 'add', 'ago', 'ou', 'mf', 'adb', 'ze', 'abx', 'xn', 'mc', 'ahf', 'wp', 'aew', 'abs', 'mg', 'wg', 'wx', 'abw', 'ma', 'vh', 'aix', 'oo', 'nt', 'wl', 'aah', 'abd', 'kx', 'aac', 'aaq', 'rd', 'wv', 'qb', 'xc', 'sf', 'sg', 'abo', 'afu', 'acm', 'aeh', 'mo', 'rg', 'yc', 'abe', 'ahr', 'mw', 'acc', 'afy', 'yr', 'ng', 'md', 'adj', 'adp', 'ahx', 'rq', 'tv', 'vn', 'ahu', 'wi', 'aij', 'uh', 'aih', 'ym', 'pp', 'ur', 'sm', 'agp', 'aad', 'wr', 'pr', 'lz', 'acv', 'te', 'ue', 'qx', 'afj', 'lh', 'nx', 'aek', 'ahc', 'yx', 'rh', 'xj', 'su', 'aal', 'aap', 'ne', 'aha', 'ls', 'tu', 'aiy', 'lu', 'aez', 'rl', 'px', 'act', 'aci', 'aim', 'acb', 'xu', 'sk', 'rm', 'aav', 'xk', 'ud', 'ms', 'ty', 'afq', 'qm', 'qd', 'ahb', 'afw', 'yz', 'ahl', 'adc', 'aex', 'mz', 'acy', 'uf', 'ow', 'zn', 'tc', 'nj', 'wt', 'ov', 'lk', 'aft', 'pe', 'wc', 're', 'yf', 'th', 'qf', 'zs', 'ni', 'mb', 'nm', 'acp', 'aeq', 'xa', 'aaw', 'ahj', 'afo', 'qq', 'agu', 'aep', 'aeu', 'mp', 'li', 'wk', 'kv', 'acw', 'yj', 'qu']\nOnly return the string element missing from Set_A.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "0sL35u", |
| "task_prompt": "You are a graduate student working in the laboratory of a formerly renowned marine natural product chemist at a well-known land-grant university. You and your colleagues have been tasked with cataloguing a historical stock of secondary metabolites that have isolated over the PI’s career and entering these into a modern chemical inventory system. The compounds have been entered into the system as small molecule input line entry system (SMILES) codes. One of these, reportedly isolated from a marine sponge, catches your attention with the following SMILES code: \n\nO[C@@H](C[C@@H](O)[C@H](OO[C@@H](CC(N[C@@H](CC(C)C)C(N[C@@H](C(N[C@H]([C@H]1C)C(N[C@@](C(C)C)([H])C(N[C@@H](CC(C)C)C(N[C@H](CO)C(N[C@@H](CC(C)C)C(N[C@H](CO)C(N[C@](CC(C)C)([H])C(O1)=O)=O)=O)=O)=O)=O)=O)=O)CCC(O)=O)=O)=O)CCCCCCCCC)CC[C@@H](O)C[C@@H](O)CC2=O)C[C@@]3(O)O[C@H](CC(O[C@@]4([H])[C@@H](OC)[C@@H](N)[C@H](OC5=CC6=C(C=C5)C(C(C7=CN(N(C8=C9C=CC(Br)=C8)C=C9C(C%10=CN(S(=O)(O)=O)C%11=C%10C=C(C%12=CC%13=C(C=C%12Br)NC=C%13C(C%14=CN(S(=O)(O)=O)C%15=C%14C=CC(Br)=C%15)=O)C(Br)=C%11)=O)C%16=C7C=CC(Br)=C%16)=O)=CN6S(=O)(O)=O)[C@@H](C)O4)/C=C/C=C/C=C/C=C/C=C(Cl)/C=C(Cl)/C=C/[C@@H](C)[C@@H](OC([C@H]%17[C@@H](CC(O[C@@]%18([H])[C@@H](OC)[C@@H](N)[C@H](O)[C@@H](C)O%18)/C=C/C=C(C%19=C%20C([C@@]%21%22C(N%20)=C(C)C[C@@]%23(CC)[C@@H]%21N(CC%22)C[C@H]%24[C@@H]%23OC%25=C%24C=C([C@@]%26%27C(N%28)=C(C(OC)=O)C[C@@]%29(CC)[C@@H]%26N(CC%27)CC=C%29)C%28=C%25)=CC([C@@H]%30[C@@H]%31[C@@H](O%31)[C@@]%32(CC)CC(C(OC)=O)=C%33[C@]%34(C(C=C(O)C(OC)=C%35OC)=C%35N%33)[C@H]%32N%30CC%34)=C%19OC)/C=C/C=C/C=C(I)/C=C(I)/C=C/[C@@H](C)[C@@H](O)[C@@H](O)[C@H](O)OC%36=O)O[C@@](C[C@@H](O)C[C@@H](O)[C@H]%37CC([C@@H](O)C[C@@H](O)C%36)=C(CCCCCCCCC%38=CC=CC=C%38)O%37)(O)C[C@@H]%17O)=O)[C@@H](O)[C@H](O)O2)[C@H](C(OC)=O)[C@@H](O)C3\n\nYou happen to have a passing interest in cheminformatics, as well as logical deduction, so you attempt to decipher the structure of the metabolite from the SMILES code. For the purpose of this exercise, determine how many hydrogen atoms there are on the molecule.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "15yKc6", |
| "task_prompt": "The receive coil of a 1.5T clinical MRI machine has a 1000 pF capacitor, soldered on one of its internal PCBs. Can it be replaced with a 2000 pF capacitor without affecting the coil's performance? We do not take into account the legal aspects of such a replacement, only the technical ones.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "1ncrzI", |
| "task_prompt": "Clinical MRI coils often consist of multiple channels, each of which is a magnetic loop. In some cases, the channels are combined into triplets, where the signals from three separate channels are combined using fixed phase shifters of 90 and 180 degrees, producing three mixed signals. What is the purpose of such a combination?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "7iy4wr", |
| "task_prompt": "In MRI gradient coil design, concomitant fields must be accounted for in the optimization process, as they can introduce artifacts into the final MR images. One of the concomitant magnetic fields along the X direction ($B_x$) can be calculated from the azimuthal and longitudinal components of the current density on the cylindrical surface. The radius ($r$) and length ($L$) of the cylinder are 0.4 m and 1.4 m, respectively. If the azimuthal ($J_\\theta$) and the longitudinal ($J_z$) components of the current density on the cylindrical surface are given by the Fourier series expansion as $J_\\theta(\\theta,z)=\\cos(\\theta)\\sum_{n=1}^{N} a_n \\cos\\!\\left(\\frac{2 n \\pi z}{L_a}\\right)$ and\n\n$J_z(\\theta,z)=\\sin(\\theta)\\sum_{n=1}^{N}\\frac{a_n L_a}{2 n \\pi r}\\sin\\!\\left(\\frac{2 n \\pi z}{L_a}\\right)$, respectively, where $L_a$ and $r$ are the length and radius of the cylinder, what will be the magnitude of the $B_x$ concomitant field at the target field point (-0.2, 0.1, 0.2) ? Assume the radial component of the current density ($J_r$)=0 and use permeability of free space ($\\mu_0$)=$4\\pi\\times 10^{-7}$ H/m. Also assume $a_n$=1 and $N=1$.\n Output your final answer in nanotesla with two decimal places. Example: 1.11 nanotesla", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "93I7Dw", |
| "task_prompt": "The receive coil of a 1.5T clinical MRI machine has a 5.6 uH inductance, soldered on one of its internal PCBs. Can it be replaced with an 8 uH inductance without affecting the coil's performance? We do not take into account the legal aspects of such a replacement, only the technical ones.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "CNODK8", |
| "task_prompt": "Let $F_2$ be the free group of rank $2$. For each $n \\geq 2$, you are given a characteristic quotient $F_2 \\to A_n$, where $A_n$ denotes the alternating group of degree $n$. This induces a map $Aut(F_2) \\to Aut(A_n)$, which you are guaranteed is surjective.\n\nFor each $n$, let $S_n$ be the direct product of the image of all complex irreducible representations of $Aut(A_n)$, and let $Aut(A_n) \\to S_n$ be the diagonal homomorphism, which is surjective on each direct factor. Composing with the above defines a homomorphism $Aut(F_2) \\to S_n$. Define\n$\\alpha_n$ as the von Neumann dimension $dim_{\\mathcal{N}(S_n)} H_0(Aut(F_2), \\mathcal{N}(S_n))$. Compute the sum of all $\\alpha_n$, for $n \\geq 2$.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "GKcq0m", |
| "task_prompt": "\nIn the following, $X=(X_n, n \\geq 0)$, $Y=(Y_n, n \\geq 0)$, $Z=(Z_n, n \\geq 0)$ etc. are \\emph{independent} discrete-time irreducible Markov chains on a countable state space $\\mathcal{X}$. Let $M^{X,Y}=\\{X_n=Y_n\\text{ for infinitely many }n\\}$ be the event that the Markov chains $X$ and $Y$ meet infinitely often; the events $M^{X,Z}$ and $M^{Y,Z}$ are defined analogously. Starting location(s) of the Markov chain(s) are indicated as subscripts; e.g., $\\mathbb{P}_{x_0,y_0}[\\cdot]$ stands for $\\mathbb{P}[\\cdot \\mid X_0=x_0, Y_0=y_0]$.\n\nClassify the following statements as true (\"True\") or false (\"False\"). Make sure you are able to rigorously prove the statements which are true and to present counter-examples (and prove that these do work) to the statements which are false. Your final answer must folow the exact format: 1. True|False 2. True|False ... 20. True|False\n\n Example:\n Answer: 1. True 2. False ... 20. True \n\n1. If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for at least one pair $x_0,y_0$, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for all $x_0,y_0$.\n\n2. If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for at least one pair $x_0,y_0$ and $X$ is aperiodic, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\n\n3. If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for at least one pair $x_0,y_0$ and both $X$ and $Y$ are aperiodic, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\n\n4. If $X$ is recurrent and $Y$ is transient, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for all $x_0,y_0$.\n\n5. If $X$ is recurrent and $Y$ is transient, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for at least one pair $x_0,y_0$.\n\n6. If both $X$ and $Y$ are recurrent, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for at least one pair $x_0,y_0$.\n\n7. For any Markov chains $X$, $Y$ and all $x_0,y_0$ it holds that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ or $1$ (i.e., we have a 0-1 law for the event $X$ and $Y$ meet infinitely often). \n\n8. If $\\mathcal{X}=\\mathbb{Z}^d$ and $X$ and $Y$ are spatially homogeneous random walks, then for all $x_0,y_0$ it holds that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ or $1$ (i.e., we have a 0-1 law for the event $X$ and $Y$ meet infinitely often). \n\n9. If $X$ and $Y$ are simple random walks on an infinite graph of uniformly bounded degree, then for all $x_0,y_0$ it holds that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ or $1$ (i.e., we have a 0-1 law for the event $X$ and $Y$ meet infinitely often). \n\n10. For any $\\alpha\\in(0,1)$ it is possible to construct an example of two Markov chains $X$, $Y$ with state space $\\mathbb{Z}$ and only nearest-neighbor jumps, such that $\\mathbb{P}_{0,0}[M^{X,Y}]=\\alpha$.\n\n11. If both $X$ and $Y$ are positive recurrent, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\n\n12. If both $X$ and $Y$ are positive recurrent and $Y$ is aperiodic, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\n\n13. If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ and $\\mathbb{P}_{x_0,z_0}[M^{X,Z}]=1$, than we have $\\mathbb{P}_{y_0,z_0}[M^{Y,Z}]=1$.\n\n14. If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=\\mathbb{P}_{x_0,z_0}[M^{X,Z}]=\\mathbb{P}_{y_0,z_0}[M^{Y,Z}]=1$ for all $x_0,y_0,z_0$, then $\\mathbb{P}_{x_1,y_1,z_1}[\\text{there exists }n>0 \\text{ such that }X_n=Y_n=Z_n]=1$ for at least one triple $(x_1,y_1,z_1)$.\n\n15. Let $X,Y$ be one-dimensional null-recurrent Markov chains with only nearest-neighbour jumps, and let $U$ be a positive recurrent Markov chain (also in $\\mathcal{X}=\\mathbb{Z}$). It is possible to construct an example such that $\\mathbb{P}_{0,0,0}[U_n=X_n=Y_n\\text{ infinitely often}]=1$.\n\n16. Let $X^{(1)},\\ldots,X^{(k)}$ be one-dimensional null-recurrent Markov chains with only nearest-neighbour jumps, and let $U$ be a positive recurrent Markov chain (also in $\\mathcal{X}=\\mathbb{Z}$). For any $k$ it is possible to construct an example such that $\\mathbb{P}_{0,\\ldots,0}[U_n=X^{(1)}_n=\\ldots=X^{(k)}_n\\text{ infinitely often}]=1$.\n\n17. Let $X,Y$ be one-dimensional Markov chains with only nearest-neighbour jumps, such that $X$ is positive recurrent and $Y$ is transient. Then $\\mathbb{P}_{0,0}[M^{X,Y}]=0$.\n\n18. Assume that the state space $\\mathcal{X} = \\mathbb{Z}^3$. It is possible to construct an example of spatially homogeneous independent random walks $X$ and $Y$ with bounded jumps such that $\\mathbb{P}_{0,0}[M^{X,Y}]=1$.\n\n19. Assume that the state space $\\mathcal{X} = \\mathbb{Z}^3$. We say that a Markov chain is \\emph{uniformly elliptic} if $\\mathbb{P}_{xy}>\\varepsilon$ for all neighbours $x,y\\in \\mathbb{Z}^3$, where $\\varepsilon>0$ is some constant. Let $X$ be a uniformly elliptic zero-mean Markov chain with uniformly bounded jumps, and $Y$ be a positive-recurrent Markov chain. Then $\\mathbb{P}_{0,0}[M^{X,Y}]=0$.\n\n20. Assume that the state space $\\mathcal{X} = \\mathbb{Z}^2$ and $X$ is a simple random walk. Then, for any uniformly elliptic zero-mean Markov chain $Y$ with uniformly bounded jumps we have $\\mathbb{P}_{0,0}[M^{X,Y}]=1$.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "ISZkG7", |
| "task_prompt": "Consider the antisymmetrized gamma matrices $\\gamma_{\\mu_1 \\ldots \\mu_k}\\equiv \\gamma_{[\\mu_1}\\ldots \\gamma_{\\mu_k]}$ in $d$ dimensions.\nThe product $\\gamma_{\\mu \\nu \\rho}\\gamma_{\\mu_1 \\ldots \\mu_k}\\gamma^{\\mu \\nu\\rho}$ is proportional to \n$\\gamma_{\\mu_1 \\ldots \\mu_k}$. What is the proportionality factor?\n Your final answer must be formatted as a LaTeX expression wrapped in $...$", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-001", |
| "task_prompt": "What is the IUPAC name of the product of methyl phenyl sulfoxide (1.0 eq.) with 1 equivalent of triflic anhydride and 1 equivalent of trimethylsilyl cyanide?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-002", |
| "task_prompt": "Aqueous solutions of barium chloride and silver nitrate were poured into one flask. The resulting mass was dried using freeze drying and ammonia was added. After mixing the reaction mixture, the ammonia was also evaporated using freeze drying. What barium salt is in the flask after all reactions? provide the molecular formula of the barium salt", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-003", |
| "task_prompt": "What is the symmetry group of the molecule with this SMILES string (C#CC1=C(C=C2)C3=C4C5=C6C7=C3C2=CC(C#C)=C7C=CC6=CC(C#C)=C5C=CC4=C1)?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-007", |
| "task_prompt": "2,8-bis(4-(2-ethylhexyl)thiophen-2-yl)-5-methyl-4H-dithieno[3,2-e:2',3'-g]isoindole-4,6(5H)-dione is an important A unit for organic solar cell polymers. We want to have bromination on the thiophene using NBS to prepare the monomer. When we added 2 eq of NBS, the spot remains the same on TLC. So we added an extra amount of NBS (0.5 eq, 2.5 eq in total). After one hour, we found a new spot on TLC. When we isolated the new spot, it showed three peaks that are larger than 6.0 ppm in H-NMR. What is this new spot?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-010", |
| "task_prompt": "Air-stable organic radicals are promising structures for OLED because it can avoid forbidden transition. But it has a strong disadvantage compared to other non-radical materials. What is that and Why? Answer Choices: A. not stable to oxygen because it is highly reactive B. wide FWDH because it has multiple emissions C. low luminance because radicals can quench each other D. low EQE because excitons can be quenched by the radicals E. low EQE because delocalization of the radical will have multiple emissions\n Your final answer must contain only an uppercase letter (A ,B ,C ,D ,E).", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-013", |
| "task_prompt": "When a student is purifying ((2R,3R,5R)-5-(4-amino-2-oxopyrimidin-1(2H)-yl)-4,4-difluoro-3-hydroxytetrahydrofuran-2-yl)methyl 2,2,2-trifluoroacetate, he used column chromatography (eluent: DCM:CH3OH = 9:1). He did a TLC and found that it contains two spots with Rf = 0.45 and Rf = 0.02, respectively. However, when he started the chromatography, nothing was coming out under these conditions. How should he solve it? Answer Choices: A. add NEt3 when he did the column B. use DCM/acetone to do the column C. wash the column before to eliminate all the water D. should not use rotovap E. repeat the experiment.\nYour final answer must contain only an uppercase letter (A ,B ,C ,D ,E).", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LLI-016", |
| "task_prompt": "Why does tributyltin chloride (TBT-Cl) tend to be less dangerous than trimethyltin chloride (TMT-Cl) for humans? Choose the most important factor. Answer Choices: A. TBT-Cl has higher boiling point for people to inhale B. TMT-Cl has a significantly lower LD50 value in mice C. TMT-Cl is more cell permeable D. TMT-Cl is more reactive to nucleophiles E. TBT-Cl can be easily degraded by human cells\nYour final answer must contain only an uppercase letter (A ,B ,C ,D ,E).", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "LbE4sy", |
| "task_prompt": "Provide a molecular formula for the depicted molecule. image location: refs/Compound_10.png", |
| "reference_file": "Compound_10.png" |
| }, |
| { |
| "task_id": "Mxadld", |
| "task_prompt": "I'm testing a clinical MRI receive coil. For this, I'm using a two-port vector network analyzer (VNA). The first channel of the VNA is connected to the magnetic loop RF probe, and the second channel is connected to the RF output of the coil. I measure S21 by placing the RF probe over the coil channel under test. What should the S21 plot shape look like if all coil components are working properly and the coil channel preamplifier is also properly powered?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "N8iZko", |
| "task_prompt": "Provide a molecular formula for the depicted molecule. image location: refs/Compound_2.png", |
| "reference_file": "Compound_2.png" |
| }, |
| { |
| "task_id": "WmZccO", |
| "task_prompt": "Consider the attached reference file which defines the bipartite graph complex induced by a graph property P and two positive integers m and n. For the following subtasks you can assume that both m and n are larger than 100 to avoid corner cases.\\n\\nT1\\nLet m be a power of 2, let n be a power of 3 and let P be the property of being triangle-free. Compute the reduced Euler characteristic of the bipartite graph complex induced by P, m, and n.\\n\\nT2\\nLet P be the property of not having a perfect matching. Let m=n be an odd prime. Compute, modulo n, the reduced Euler characteristic of the bipartite graph complex induced by P, m, and n.\\n\\nT3\\nLet P be the property of being planar. Let m = n-1. Compute, modulo n, the reduced Euler characteristic of the bipartite graph complex induced by P, m, and n.\\n\\nT4\\nLet P be the property of not containing as a subgraph the complete bipartite graph with 101 vertices on each side. Let m=1000, and let n be a power of 23. Compute, modulo 23, the reduced Euler characteristic of the bipartite graph complex induced by P, m, and n.\\n\\nT5\\nLet P be the property of having treewidth smaller than 5, and let n be a power of a prime q. Let furthermore m<n. Compute, modulo q, the reduced Euler characteristic of the bipartite graph complex induced by P, m, and n.\\n\\nT6\\nLet n=m be prime. Define two graph properties P and Q to be equivalent if they agree on all bipartite graphs. Compute, as a function of n, the number of equivalence classes [P] such that the reduced Euler characteristic of the bipartite graph complex induced by P, m, and n is divisible by n.\\n", |
| "reference_file": "main.pdf" |
| }, |
| { |
| "task_id": "XE4oUN", |
| "task_prompt": "Let $H$ a three-form, $\\alpha_\\pm$ an even/odd form. Consider the Clifford map $C(d x^{\\mu_1} \\wedge \\ldots \\wedge d x^{\\mu_k}):= \\gamma^{\\mu_1\\ldots \\mu_k}$. \n\nFind $c_{1,\\ldots,4}$ such that\n$ C(H \\wedge \\alpha_\\pm) =\nc_1 C(H) C(\\alpha_\\pm) + c_2 C(\\alpha_\\pm) C(H) +c_3 \\gamma^\\mu C(\\alpha_\\pm) C(\\iota_\\mu H) + c_4 C(\\iota_\\mu H) C(\\alpha_\\pm) \\gamma^\\mu$.\n Your final answer must be in the following format: $c_1=...$, $c_2=...$, $c_3=...$, $c_4=...$", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "eXEvuO", |
| "task_prompt": "A graduate student wants to design a biplanar MRI Z-gradient coil for a C-shaped low-field scanner. For the design, two circular discs, each having a radius of 0.4 and located at z=0.3 and z=-0.3 planes, were used. The surface current densities ($J_\\theta$) of both discs can be approximated using the Fourier series expansion as : \n$J_\\theta(\\rho)=\\sum_{n=1}^{N} a_n \\sin\\!\\left(\\frac{n\\pi(\\rho-\\rho_0)}{\\rho_m-\\rho_0}\\right), \\quad \\text{where } \\rho_m=0.4, \\text{and}\\ \\rho_0=0.1$ are the maximum and minimum radii of the discs, respectively. What will be the magnitude of the longitudinal component of the magnetic field ($B_z$($x_f$, $y_f$, $z_f$)) at the target point (0.1, 0.1, -0.1)? Use permeability of free space ($\\mu_0$)=$4\\pi\\times 10^{-7}$ H/m and assume all measurements are in SI units. Also, assume $a_n=1$ and $N=1$.\n Output your final answer in nanotesla at two decimal places. Example: 1.11 nanotesla", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "electron-microscopy_0b02fbe1", |
| "task_prompt": "refs/619e2c2c.png is an image of a crop from a thin section electron microscopy image of flagellate eukaryote. Which flagellar structure is the main feature of the image? \n Options: basal body, transition zone, axoneme, axoneme with paraxial structure. Briefly justify your choice using visible structural features, then on the last line provide only one option from the list.", |
| "reference_file": "619e2c2c.png" |
| }, |
| { |
| "task_id": "electron-microscopy_27300012", |
| "task_prompt": "refs/0bde932a.png is an image of a crop from a thin section electron microscopy image of a mammalian cell. Which organelle from the following options is the main feature of the image? \n Options: nucleolus, nucleolus, nuclear envelope, cytoplasm, rough endoplasmic reticulum, microtubule, actin filament, clathrin-coated pit, vesicle, mitochondrion. Briefly justify your choice using visible structural features, then on the last line provide only one option from the list.", |
| "reference_file": "0bde932a.png" |
| }, |
| { |
| "task_id": "electron-microscopy_3cb48fbe", |
| "task_prompt": "The Leishmania parasite has a single flagellum which loses the capacity for motility in the amastigote, and the axoneme collapses into a non-circular organisaton lacking a central pair termed 9v. This series of cropped thin section electron microscopy images shows cross-sections through a Leishmania amastigote flagellum at different positions along its length. The images are shown in order from top to bottom of column one then column two. It would be expected that the more circular arrangement is near the base, near the circular basal body, but the images do show the basal body to directly confirm that. Without making that assumption, informed by chirality of the axoneme structure, state where the collapse occurs: A. Towards the base. B. Towards the tip. C. Cannot be determined. Briefly justify your choice using the visible axoneme chirality and the information given, then on the last line provide only the letter corresponding to the answer from the list.", |
| "reference_file": "0a63453b.png" |
| }, |
| { |
| "task_id": "electron-microscopy_433becb7", |
| "task_prompt": "refs/85a0e4f5.png is an image of a crop from a thin section electron microscopy image showing part of the flagellum structure. In which direction is the view facing? A. Towards the flagellum base. B. Towards the flagellum tip. C. Depends on the species. D. Cannot be determined. Briefly justify your choice using the structural chirality visible in the image, then on the last line provide only the letter corresponding to the answer from the list.", |
| "reference_file": "85a0e4f5.png" |
| }, |
| { |
| "task_id": "electron-microscopy_48ab3083", |
| "task_prompt": "Trichodina is a genus of ciliated ectoparasite that attach to the surface of fish and other aquatic organisms. They have a cilia-lined feeding groove which leads to the cytostome. refs/a66b1e37.png is a thin section electron microscopy image of the feeding groove. Based on this image, is the micograph looking down into the feeding groove or up out of the feeding groove? A. Looking down into the feeding groove. B. Looking up out of the feeding groove. C. Cannot be determined from the image. Briefly justify your choice using visible structural features, then on the last line provide only the letter corresponding to the answer from the list.", |
| "reference_file": "a66b1e37.png" |
| }, |
| { |
| "task_id": "electron-microscopy_97c56090", |
| "task_prompt": "refs/d465ab9a.png is an image of a crop from a thin section electron microscopy image of the Leishmania parasite. The Leishmania parasite swims using a single flagellum which has an asymmetric paraxial structure with a lattice-like internal structure, the PFR, as shown in the thin section electron microscopy image on the on the left. An axoneme mutation leads to curling up of the flagellum, as shown in the scanning electron microscopy of a detergent-extracted cell on the right, with the axoneme and PFR visible. The preferred bending plane of the flagellum is determind by the central pair microtubules. Dynein motor proteins on the axoneme doublets cause microtubule sliding which leads to flagellum bending. Based on these images, the dyneins on which set of doublets are over-active, leading to the curl? A. Doublets 1, 2, 3, 8 and 9. B. Doublets 1, 2, 3 and 4. C. Doublets 3, 4, 5, 6, 7. D. Doublets 6, 7, 8, 9. E. Cannot be determined from the images. Briefly justify your choice using the visible bend direction and axoneme geometry, then on the last line provide only the letter corresponding to the answer from the list.", |
| "reference_file": "d465ab9a.png" |
| }, |
| { |
| "task_id": "hUkto8", |
| "task_prompt": "Provide a molecular formula for the depicted molecule. image location: refs/Compound_6.png", |
| "reference_file": "Compound_6.png" |
| }, |
| { |
| "task_id": "hv9qc0", |
| "task_prompt": "Consider the antisymmetrized gamma matrices $\\gamma_{\\mu_1 \\ldots \\mu_k}\\equiv \\gamma_{[\\mu_1}\\ldots \\gamma_{\\mu_k]}$ in $d$ dimensions.\nThe product $\\gamma_{\\mu \\nu \\rho \\sigma}\\gamma_{\\mu_1 \\ldots \\mu_k}\\gamma^{\\mu \\nu\\rho \\sigma}$ is proportional to \n$\\gamma_{\\mu_1 \\ldots \\mu_k}$. What is the proportionality factor?\n Your final answer must be formatted as a LaTeX expression wrapped in $...$", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "il0nMv", |
| "task_prompt": "Determine the molecular formula of the following molecule: CC[C@H](C)[C@@H](C(=O)N[C@H](CC(=O)NC)C(=O)N[C@@H](C(C)C)C(=O)N[C@H]([C@H](C(=O)NC)O)C(=O)N[C@@H](C)C(=O)N[C@H](CC(=O)NC)C(=O)N[C@@H](C(C)C)C(=O)N[C@H](CO)C(=O)N[C@@H](C(C)C)C(=O)N[C@H](CC(=O)N)C(=O)N[C@H](C(=O)N[C@H](CC(=O)N)C(=O)N[C@@H](CCC(=O)N)C(=O)N[C@H]([C@@H](C)O)C(=O)N[C@@H]([C@@H](C)O)C(=O)O)C(C)(C)C[S@](=O)C)NC(=O)[C@@H](CC(=O)NC)NC(=O)CNC(=O)[C@@H](C(C)(C)O)NC(=O)[C@H](C(C)(C)C)NC(=O)[C@@H]([C@H](C(=O)NC)O)NC(=O)[C@H]([C@@H](C)CC)NC(=O)[C@@H](CC(=O)NC)NC(=O)CNC(=O)CNC(=O)[C@H](C)NC(=O)[C@@H](C(C)(C)O)NC(=O)[C@H]([C@@H](C)CC(=O)N)NC(=O)[C@@H](CC(=O)NC)NC(=O)[C@H](C(C)(C)C)NC(=O)CNC(=O)[C@H](C)NC(=O)CNC(=O)[C@H](C(C)(C)O)NC(=O)[C@@H](CC(=O)NC)NC(=O)[C@H](C)NC(=O)[C@@H](C(C)(C)C)NC(=O)[C@H](C)NC(=O)CNC(=O)[C@H](C)NC(=O)[C@@H](C(C)(C)C)NC(=O)[C@H](C(C)(C)C)NC(=O)[C@@H](C)NC(=O)[C@H](C(C)(C)C)NC(=O)[C@@H](C(C)(C)C)NC(=O)[C@H](C(C)(C)C)NC(=O)CNC(=O)[C@H](C(C)(C)CC)NC(=O)CNC(=O)C(=O)CCC(C)(C)C", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "iysdcv", |
| "task_prompt": "A transportation concept consists of a capsule travelling inside a partially evacuated tube that is closed at both ends with total length 570 km. The capsule has a cross-sectional area of $A_{\\\\\\\\text{capsule}} = 4$ m$^2$. You should assume that the capsule is a `tight fit' inside the tube (i.e., the capsule cross-sectional area is approximately equal to the tube cross-sectional area). The initial air pressure and temperature inside the tube are 100 Pa and 300 K, respectively. Air may be treated as a perfect gas with $R = 287$ J kg$^{-1}$ K$^{-1}$ and $\\\\\\\\gamma = 1.4$.\\\\n\\\\nAt time $t = 0$, an initially stationary capsule starts to accelerate at 10 m s$^{-2}$ away from its initial position, which is exactly half way along the tube. At $t = 34$ s, the capsule stops accelerating and maintains a constant speed $V = 340$ m s$^{-1}$.\\\\n\\\\nCalculate the power required to propel the capsule at $V = 340$ m s$^{-1}$ shortly after it stops accelerating (at $t\\\\\\\\approx 40$ s). Provide your answer to the nearest kW.\\\\n", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "khs9kS", |
| "task_prompt": "Define $c$ such that $\\gamma\\equiv c\\,\\gamma^1\\ldots \\gamma^d$, with $\\gamma$ the chiral operator (so $d$ is even). Define $\\lambda$ such that $\\lambda \\alpha_k \\equiv (-1)^{\\lfloor \\frac k2 \\rfloor} \\alpha_k = (-1)^{\\frac{k(k-1)}2} \\alpha_k $, where $\\alpha_k$ is a $k$-form. Consider the Clifford map $C(d x^{\\mu_1} \\wedge \\ldots \\wedge d x^{\\mu_k}):= \\gamma^{\\mu_1\\ldots \\mu_k}$. \n\nFind $p$ such that $\\gamma C(\\alpha) = p C(* \\lambda \\alpha)$.\n Your final answer must be formatted as a LaTeX expression wrapped in $...$", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "maBY4w", |
| "task_prompt": "Biot-Savart's law is essential for calculating the magnetic field at any desired point during Magnetic Resonance Imaging (MRI) gradient coil design. Biplanar configurations are usually used for the design of gradient coils for C-shaped scanners. The surface current densities ($J$) of two circular discs at z=-0.2m and z=0.2 m, respectively, were used to generate the desired gradient field at the target field points. The radii of both plates are 0.3. The azimuthal (J_{\\theta}) and radial ($J_r)$ surface current densities of both the first and second discs are approximated by the Fourier series expansion as: \n$J_\\rho(\\rho,\\theta)=\\sum_{n=1}^{N} \\frac{a_n}{\\rho} \\sin\\big(n c (\\rho-\\rho_0)\\big) \\sin(\\theta)$ and\n\n$J_\\theta(\\rho,\\theta)=\\sum_{n=1}^{N} n c a_n \\cos\\big(n c (\\rho-\\rho_0)\\big) \\cos(\\theta), \\quad \\text{where } c=\\frac{\\pi}{\\rho_m-\\rho_0}, \\ \\rho_m=0.3,\\text{and} \\ \\rho_0=0.1$ are the maximum and minimum radii of the discs, respectively.\nWhat will be the magnitude of the longitudinal component of the magnetic field ($B_z$($x_f$, $y_f$, $z_f$)) at the target point (0.1, 0.1, -0.1)? Use permeability of free space ($\\mu_0$)=$4\\pi\\times 10^{-7}$ H/m and assume all measurements are in SI units. Also, assume $N=1$ and $a_n=1$.\n Output your final answer in micro Tesla with three decimal places. Example: 1.111 microtesla", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "pQ18l0", |
| "task_prompt": "You are a marine natural products chemistry post-doc within the Chinese academic system hoping to apply for a tenured position within the coming year, as such, your time is extremely valuable. You are exploring marine sponges of the Latrunculiidae family and have isolated a compound of interest for which you are able to obtain a (dilute) 1H NMR spectrum. Spectral data for the species with chemical shift, multiplicity, integral and coupling constant is as follows: 1H NMR (methanol-d4, 500 MHz) δ [ppm] 7.15 (s, 1H), 7.07 (d, 2H, J = 8.5 Hz), 6.72 (d, 2H, J = 8.5 Hz), 5.39 (s, 1H), 3.86, (bt, 2H, J = 7.5 Hz), 3.54 (t, 2H, J = 7.5 Hz), 2.94 (t, 2H, J = 7.5 Hz), 2.87 (t, 2H, J = 7.5 Hz). You suspect that this may be a known compound, but cannot be certain. Using your extensive knowledge of chemotaxonomy and the literature surrounding proton chemical shifts, you have attempted to dereplicate the structure of the isolated compound. For the purpose of answering this question, provide a molecular formula for the compound.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_108", |
| "task_prompt": "Consider the transverse-field Ising chain with boundary terms as\n\\begin{equation}\n\tH = - \\sum_{j=1}^{N} ( \\sigma_j^x \\sigma_{j+1}^x + g \\sigma_j^z ) + h_b \\sigma_N^x \\sigma_1^x - h_1 \\sigma_1^x\n\\end{equation}\nwhere $\\sigma_j^{x,y,z}$ are Pauli matrices on site $j$, $N$ is even, and $\\sigma_{N+1} \\equiv \\sigma_1$.\nDefine\n\\begin{equation}\n\t\\mathrm{D} = e^{-\\frac{2\\pi i N}{8}} U \\frac{1 + \\eta}{2} , \\quad\n\tU = \\prod_{j=1}^{N-1} \\left( \\frac{1 + i \\sigma^z_j}{\\sqrt{2}} \\frac{1 + i \\sigma^x_j \\sigma^x_{j+1}}{\\sqrt{2}} \\right) \\left( \\frac{1 + \\sigma^z_N}{\\sqrt{2}} \\right) \\left( \\frac{1 - \\sigma^x_1}{\\sqrt{2}} \\right) \\left( \\frac{1 - \\sigma^z_N}{\\sqrt{2}} \\right) ,\n\\end{equation}\nwhere $\\eta = \\prod_{j=1}^{N} \\sigma^z_j$, and the product over $j$ is in an ascending order.\nEvaluate\n\\begin{equation}\n\t| \\langle \\psi | \\mathrm{D} | \\psi \\rangle |\n\\end{equation}\nwhere $|\\psi\\rangle$ is the ground state of Hamiltonian $H$ at $g = 1, h_b = 0, h_1 = 0$.\n Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_110", |
| "task_prompt": "Consider a non-local matrix product operator that constructs a quantum circuit, defined as\n\\begin{equation}\nO = U \\left( \\sqrt{2} - X_N - Z_N \\right) U^\\dagger , \\quad\nU = \\prod_{j=1}^{N-2} \\left( \\frac{1 + Z_j X_{j+1} Z_{j+2}}{\\sqrt{2}} \\frac{1 + Y_j \\mathrm{CNOT}_{j, j+1}}{\\sqrt{2}} \\right) \\left( \\frac{1 + H_N}{\\sqrt{2}} \\right) ,\n\\end{equation}\nwhere $X_j, Y_j, Z_j$ are Pauli matrices on site $j$, $H_j$ is the Hadamard gate on site $j$, $\\mathrm{CNOT}_{j, j+1}$ is the CNOT gate on sites $j, j+1$, and the product over $j$ is in an ascending order. Evaluate the bond dimension of the matrix product operator $O$.\n Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_117", |
| "task_prompt": "In a two–site impurity model at half‐filling and possessing particle–hole symmetry, the retarded impurity Green’s function in the time domain is defined as\n\\[\nG^{R}_{\\text{imp}}(t)= -i\\,\\theta(t)\\,\\langle\\{c(t),c^{\\dagger}(0)\\}\\rangle ,\n\\]\nwith $\\theta(t)$ the Heaviside step function.\nBecause the spectrum is composed of two symmetric pairs of excitations at the energies $\\pm\\omega_{1}$ and $\\pm\\omega_{2}$, the Green’s function can be written as\n\\[\niG^{R}_{\\text{imp}}(t)=2\\!\\left(\\alpha_{1}\\cos(\\omega_{1}t)+\\alpha_{2}\\cos(\\omega_{2}t)\\right),\n\\]\nwhere $\\alpha_{1}$ and $\\alpha_{2}$ are the (positive) spectral weights carried by the two pairs, respectively.\n\nThe corresponding spectral function therefore reads\n\\[\nA(\\omega)=2\\alpha_{1}\\!\\left[\\delta(\\omega-\\omega_{1})+\\delta(\\omega+\\omega_{1})\\right]+2\\alpha_{2}\\!\\left[\\delta(\\omega-\\omega_{2})+\\delta(\\omega+\\omega_{2})\\right].\n\\]\n\nThe exact sum rule for a single–impurity problem imposes\n\\[\n\\int_{-\\infty}^{\\infty}A(\\omega)\\,d\\omega=1 .\n\\]\n\nExperiments give the spectral weight of the first symmetric pair as $\\alpha_{1}=0.15$.\n\nDetermine the spectral weight $\\alpha_{2}$ of the second symmetric pair $\\pm\\omega_{2}$, rounded to two decimal places. Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_120", |
| "task_prompt": "Consider the tripartite qutrit Hilbert space\n\\[\n\\mathcal{H}=\\mathcal{H}_A\\otimes\\mathcal{H}_B\\otimes\\mathcal{H}_C,\\qquad\n\\mathcal{H}_i\\cong\\mathbb{C}^3 .\n\\]\n\nDefine the pure GHZ–like state\n\\[\n\\ket{\\Psi}=\\frac{1}{\\sqrt{3}}\\bigl(\\ket{000}+\\ket{111}+\\ket{222}\\bigr),\n\\]\nand the mixed state\n\\[\n\\rho=\\frac14\\ket{000}\\bra{000}+\\frac34\\ket{\\Psi}\\bra{\\Psi}.\n\\]\n\nA noisy channel acts only on subsystem $A$:\n\\[\nE_1(\\rho)=\\frac14\\rho+\\frac34\\,X_1\\rho X_1 ,\\qquad\nX_1=X\\otimes I\\otimes I ,\n\\]\nwhere the qutrit “bit–flip’’ $X$ obeys\n\\[\nX\\ket0=\\ket1,\\quad X\\ket1=\\ket2,\\quad X\\ket2=\\ket0 .\n\\]\n\nLet\n\\[\n\\rho_{AB}'=(E_1\\otimes I_{BC})(\\rho)\n\\]\nbe the post–channel joint state of $A$ and $B$. The Rényi entropy of order $\\alpha=\\tfrac12$ is defined as\n\\[\nS^{1/2}(\\sigma)=\\frac{1}{1-\\alpha}\\,\\log\\!\\bigl(\\operatorname{Tr}\\sigma^{\\alpha}\\bigr)\n =2\\log\\!\\Bigl(\\sum_j\\sqrt{\\lambda_j}\\Bigr),\n\\]\nwith $\\{\\lambda_j\\}$ the eigenvalues of $\\sigma$.\n\nTask: Compute the order–$1/2$ Rényi \\emph{conditional} entropy\n\\[\nS^{1/2}(A|B)=S^{1/2}(\\rho_{AB}')-S^{1/2}(\\rho_B),\n\\qquad\n\\rho_B=\\operatorname{Tr}_{A}(\\rho_{AB}),\n\\]\nand give its numerical value (to three decimal places, natural logarithms understood).\nShow your reasoning, then give the final numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_138", |
| "task_prompt": "A GaAs/AlGaAs quantum cascade detector (QCD) is embedded in a symmetric double-metal (Au–GaAs–Au) Fabry–Pérot micro-cavity that can be modeled with temporal coupled-mode theory (CMT).\n1. Empty–cavity characterization (no quantum wells, i.e.\\ $P=0$) under normal incidence gives\n • minimum reflectivity $R_{\\min}=0.2$,\n • spectral full-width at half-maximum (FWHM) $\\Delta\\omega = 30\\;\\text{meV}$.\n\n\n2. After inserting the quantum-well active region, the inter-subband (ISB) transition couples to the cavity mode with\n • inter-subband decay rate $\\gamma_P=\\gamma_{\\text{ISB}}=10\\;\\text{meV}$,\n • vacuum Rabi frequency $\\Omega_R=4\\;\\text{meV}$.\n\n\n\nAssuming perfect triple resonance $\\omega=\\omega_a=\\omega_P$, use the data above to:\n\nCalculate the inter-subband absorption efficiency $\\eta_{\\text{ISB}}$ of the detector at resonance, rounded to two decimal places. Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_14", |
| "task_prompt": "Let $\\ket{0}$ ($n\\in\\mathbb{N}$) be the vacuum state of a bosonic mode, $\\hat{a}$ be the annihilation operator, and $\\ket{z}$ be the eigenstate of $\\hat{a}$ with the eigenvalue $z\\in\\mathcal{C}$. Please find the expression of $\\bra{z} \\left(\\hat{a}^{\\dagger 2} \\hat{a}^{3}\\right)^{4}\\ket{z}$ in terms of $z$. (Solving this problem with SymPy or QuTiP package in Python is prohibited.)\nShow your reasoning.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_148", |
| "task_prompt": "The action of the $N=4$ SYM model in $d=3+1$ reads\n\n$$\n\\mathcal{L}=-\\frac{1}{8}\\left(D_{A \\dot{B}} \\phi^{I J, a}\\right)\\left(D^{A \\dot{B}} \\phi_{I J}^a\\right)+i \\lambda_A^{I, a} D^{A \\dot{B}} \\bar{\\lambda}_{\\dot{B} I, a}-\\frac{1}{4} F_{A B}^a F^{A B, a}+\\mathcal{L}_{\\text {Yuk }}(S U(4))+\\mathcal{L}_{D+F}(S U(4)),\n$$\n\nwhere\n\n$$\n\\begin{gathered}\n\\mathcal{L}_{\\mathrm{Yuk}}(S U(4))=k_{\\mathrm{Yuk}} f_{a b c} \\phi_{I J}^a \\lambda^{b I A} \\lambda_A^{c J}+c . c ., \\\\\n\\mathcal{L}_{F+D}(S U(4))=k_{D+F}\\left(f_{a b c} \\phi_{I J}^b \\phi_{K L}^c\\right)\\left(f_{a b^{\\prime} c^{\\prime}} \\phi^{I J, b^{\\prime}} \\phi^{K L, c^{\\prime}}\\right)\n\\end{gathered}\n$$\n\n\nWe want to fix the coefficients $k_{D+F}$, given the Yukawa and $\\phi^4$ terms of the coupling of the $N=1$ SYM multiplet $\\left(A_\\mu^a, \\lambda_A^a, D^a\\right)$ to three WZ multiplets $\\left(\\phi^{a i}, \\psi_A^{a i}, f^{a i}\\right)$ with $i=1,2,3$. And $k_{\\text {Yuk }}=\\frac{\\sqrt{2}}{2}$.\nGiven the $D$-term in the SYM - $(\\mathrm{WZ})^3$ coupling\n\n$$\n\\mathcal{L}_D=\\frac{1}{2}\\left(f_{a b c} \\varphi_i^{* b} \\varphi^{i c}\\right)^2\n$$\n\nfix $k_{D+F}$ by matching $\\mathcal{L}_D$ to the corresponding terms in $\\mathcal{L}_{F+D}(S U(4))$.\nShow your reasoning, then give the final value of k_{D+F}.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_150", |
| "task_prompt": "Consider the action as a sum of\n\n$$\n\\mathcal{L}_{\\mathrm{HE}}=-\\frac{1}{4 \\kappa^2} \\epsilon^{\\mu \\nu \\rho} \\epsilon_{m n r} R_{\\mu \\nu}{ }^{m n}(\\omega) e_\\rho{ }^r\n$$\nand the gravitino action $\\mathcal{L}_{\\mathrm{RS}}$, replacing $e\\gamma^{\\mu \\rho \\sigma}$ in $\\mathcal{L}_{\\mathrm{RS}}$ by $B \\epsilon^{\\mu \\rho \\sigma} I$. Determine $B$. Note that the Rarita-Schwinger action also takes the same expression as in $d=4$.\n Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_24", |
| "task_prompt": "A modified amplitude damping channel $\\mathcal{A}$ has an operator-sum representation $A\\sim\\{K_1,K_2\\}$. $K_i$ is the Kraus operators defined as follows:\n\n$$\nK_1 = \\begin{bmatrix}\n 1&0\\\\\n 0&e^{i\\phi}\\sqrt{1-p}\n\\end{bmatrix},\\quad\nK_1 = \\begin{bmatrix}\n 0&\\sqrt{p}\\\\\n 0&0\n\\end{bmatrix}\\;,\n$$\nwhere $p\\in[0,1]$ is the damping parameter and $\\phi\\in[0,2\\pi)$ is some phase shift.\nThe Choi matrix of a linear map $\\Phi$ is denoted as $J(\\Phi):\\mathcal{H}\\otimes\\mathcal{H}\\to \\mathcal{H}\\otimes\\mathcal{H}$, where $\\mathcal{H}=\\mathbb{C}^2$ is a Hilbert space.\nWe also define the swap operator $S$ for the bipartite system $\\mathcal{H}\\otimes\\mathcal{H}$, such that for any $\\ket{f}\\otimes \\ket{g}\\in\\mathcal{H}\\otimes\\mathcal{H}$, we have $S\\ket{f}\\otimes \\ket{g} = \\ket{g}\\otimes \\ket{f}$.\n\nLet $\\ket{+} :=\\frac{1}{\\sqrt{2}}\\left(\\ket{0}+\\ket{1}\\right)$ and\n\n$$\n\\begin{aligned}\nX &:= J(\\mathcal{A})\\otimes S \\otimes J(\\mathcal{A})\\otimes S \\otimes J(\\mathcal{A})\\\\\nY &:= \\ket{+}\\!\\bra{+}\\otimes J(I)\\otimes J(I)\\otimes J(I)\\otimes J(I)\\otimes \\ket{+}\\!\\bra{+}\n\\end{aligned}\n$$\n\nPlease find the expression of $\\operatorname{tr}(XY)$, in terms of $\\phi$ and $p$.\nShow your reasoning, then give the final expression.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_25", |
| "task_prompt": "Let $G = (V,E)$ be a simple undirected graph, where the vertex set $V = A_1\\cup A_2 \\cup A_3$.\nThere is no edge connecting vertices within each $A_i$.\nThe vertices in $A_i$ are listed as follows:\n$$\nA_1 = \\{v_1,v_2,v_3\\},\\quad\nA_2 = \\{v_7,v_8,v_9,v_{10},v_{11},v_{12}\\},\\quad\nA_3 = \\{v_4,v_5,v_6\\}\n$$\nThe biadjacency matrix between $A_1$ and $A_2$ is denoted as $B^{(1,2)}$. The biadjacency matrix between $A_2$ and $A_3$ is denoted as $B^{(2,3)}$.\n$$\nB^{(1,2)} = \\begin{bmatrix}\n 1&1&1&1&1&1\\\\\n 1&0&1&1&1&1\\\\\n 1&1&1&1&1&1\n\\end{bmatrix},\\quad\nB^{(2,3)} = \\begin{bmatrix}\n 1&1&1\\\\\n 1&1&1\\\\\n 1&0&1\\\\\n 1&1&1\\\\\n 1&1&1\\\\\n 1&1&1\n\\end{bmatrix}\n$$\nWe consider a special quantum graph state $\\ket{G}$ living on the graph $G$, such that each qubit corresponds to a vertex in $V$.\n\nNow let us perform $\\ket{+}\\bra{+}_i$ projection on the following qubits $i=7,8,9,\\cdots,12$. The projection outcome is then normalized as a pure state $\\ket{\\psi}$.\nPlease find the mutual information $I(A_1:A_3)=S(A_1) + S(A_3) - S(A_1\\cup A_3)$ for the pure state $\\ket{\\psi}$.\n Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_26", |
| "task_prompt": "Consider a completely-positive trace-preserving map $\\mathcal{E}_\\theta$ for a bosonic mode, which encodes the parameter $\\theta$ in the following way:\n$$\n\\rho(\\theta) = \\mathcal{E}_\\theta(\\rho(0)) = \\frac{1}{\\sqrt{2\\pi \\theta}}\\int_{-\\infty}^{\\infty} e^{-\\frac{x^2}{2\\theta}} e^{-ix\\hat{n}}\\rho(0) e^{ix\\hat{n}}\\;\\mathrm{d}x\\;,\n$$\nwhere $\\hat{n}$ is the particle number operator, and $\\rho(0)$ is the input state. Now we consider $\\rho(0) = \\ket{\\alpha, r}\\bra{\\alpha, r}$ as the density matrix of the squeezed coherent state, which is defined as follows\n$$\n\\ket{{\\alpha, r}} = e^{\\alpha a^\\dagger - \\alpha^* a} e^{\\frac{r}{2}(a^2 - a^{\\dagger 2})}\\ket{0}\\;,\n$$\nwhere $\\ket{0}$ is the vacuum state, $a$ is the annihilation operator.\nPlease find the quantum Fisher information (based on symmetric logarithmic derivative) at $\\theta\\to 0$ in terms of $\\alpha\\in\\mathbb{C}$ and $r\\in\\mathbb{R}$.\nSolving this problem with SymPy or QuTiP package in Python is prohibited and will be marked as wrong.\nShow your reasoning.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_29", |
| "task_prompt": "Hypergraph product (HGP) codes have many promising features in modern quantum error correction, which has attracted significant research attention. Consider an HGP code $\\mathcal{C}$ with parameter $[[13,1,3]]$ generated by two $[3,1,3]$ Hamming codes (3-bit repetition codes).\nThe bit string $s$ obtained from the stabilizer check measurements is called the syndrome.\nA nonzero syndrome $s$ can typically be caused by a set of different Pauli errors\n$$\nA(s):=\\{E: \\text{the syndrome of error } E \\text{ is } s\\}.\n$$\nSince the decoder should determine the unique error from a given $s$. We are interested in the set of nice syndromes\n$$\nD=\\{s: A(s) \\text{ has a unique element of minimal weight}\\}.\n$$\nFor every $s\\in D$ , the minimum-weight Pauli error $E_s^{\\text{wt}_{\\min}}\\in A(s)$ is found. We denote $M_{\\mathcal{C}}=\\{E_s^{\\text{wt}_{\\min}}:s\\in D\\}$ as the set of Pauli operators that cause nonzero syndromes. What is the maximal weight of Pauli errors in $M_{\\mathcal{C}}$?\n Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_3", |
| "task_prompt": "The 5-qubit Rains code is usually denoted as the $((5,6,2))$ quantum code. Let $\\mathcal{P}_5/U(1)$ be the set of 5-qubit Pauli errors, $\\Pi_{\\mathcal{C}}$ be the projector of the code space, and $\\Pi_{\\mathcal{C},E}':=E\\Pi_{\\mathcal{C}}E$ with $E\\in \\mathcal{P}_5/U(1)$. How many $E$'s are there satisfying the orthogonal relationship $\\Pi_{\\mathcal{C},E}'\\Pi_{\\mathcal{C}} = 0$ and have weight $2$?\n Show your reasoning, then on the last line provide only the numerical value.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_58", |
| "task_prompt": "Define\n$$\nA_N^{(0)}(\\text { open })=C_N g^{N-2} \\int_{\\substack{0 \\leq z_2 \\leq \\cdots \\leq z_{N-2} \\leq 1 ; \\\\ z_N \\rightarrow \\infty, z_{N-1}=1, z_1=0}} d z_2 \\cdots d z_{N-2} z_N^2\\langle 0| \\mathcal{T} V\\left(k_N, z_N\\right) \\cdots V\\left(k_1, z_1\\right)|0\\rangle(2 \\pi)^{26} \\delta^{26}\\left(\\sum_{j=1}^N k_j\\right)\n$$\nThe following relation holds on-shell for tree graphs\n$$\n\\int A_3^{(0)}\\left(k_1, k_2, k\\right) \\frac{-i}{k^2+m^2-i \\epsilon} A_3^{(0)}\\left(k, k_3, k_4\\right) \\frac{d^{26} k}{(2 \\pi)^{26}}=\\left.A_4^{(0)}\\left(k_1, k_2, k_3, k_4\\right)\\right|_{\\text {at tachyon pole }}\n$$\n\nConsider the open string. $C_4$ can be expressed in terms of $C_3$. If $C_4=C_3$, what is the value of $C_3$? Here, we used $\\frac{l^2}{2}$ instead of $\\alpha'$.\nShow your reasoning, then give the final value of C_3.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_60", |
| "task_prompt": "Consider the open string amplitude\n\n$$\nA_N^{(0)}(\\text { open })=C_N g^{N-2} \\int_{\\substack{0 \\leq z_2 \\leq \\cdots \\leq z_{N-2} \\leq 1 ; \\\\ z_N \\rightarrow \\infty, z_{N-1}=1, z_1=0}} d z_2 \\cdots d z_{N-2} z_N^2\\langle 0| \\mathcal{T} V\\left(k_N, z_N\\right) \\cdots V\\left(k_1, z_1\\right)|0\\rangle(2 \\pi)^{26} \\delta^{26}\\left(\\sum_{j=1}^N k_j\\right)\n$$\nOne can define the corresponding closed string amplitude with $C_N$ straightforwardly. Express $C_4$ in terms of $C_3$ by looking at the pole of the four point amplitude. Suppose that $\\frac{1}{2} l^2 k^2=4$ instead of using $\\alpha'k^2=4$. What is the value of $C_3$?\nShow your reasoning, then give the final value of C_3.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prob_70", |
| "task_prompt": "Consider the Choi state $J(\\Lambda)$ of a quantum channel $\\Lambda:\\mathcal{L}(\\mathcal{H}_1)\\to \\mathcal{L}(\\mathcal{H}_2)$, where $\\mathcal{H}_1$ is the Hilbert space for a qubit and $\\mathcal{H}_2$ is an infinite-dimensional Hilbert space. $\\mathcal{L}(\\mathcal{H}):=\\{A|A: \\mathcal{H} \\to \\mathcal{H}\\}$.\n$J(\\Lambda)$ has the following matrix representation acting on $\\mathcal{H}_2\\otimes \\mathcal{H}_1$ as follows:\n\n$$\nJ(\\Lambda) = \\operatorname{Diag}\\left(A_1,A_2,A_3,\\cdots \\right)\\;,\n$$\nwhere $A_n: \\mathbb{R}^{2\\time 2}$ with parameter $\\gamma := \\arctan\\sqrt{15}$:\n\n$$\nA_n = \\frac{1}{2^n}\\begin{bmatrix}\n 1 & \\frac{1}{n}\\cos\\left(\\gamma n\\right)\\\\\n \\frac{1}{n}\\cos\\left(\\gamma n\\right) & 1\n\\end{bmatrix}\\;.\n$$\nLet $\\tau=\\frac{I_2}{2}$ be a maximally mixed state acting on $H_1$, and $\\Lambda^c$ be the complementary channel of $\\Lambda$, and $\\mathcal{R}: \\mathcal{L}(\\mathcal{H}_2)\\to \\mathcal{L}(\\mathcal{H}_1)$ be a recovery map (completely positive and trace non-increasing).\nPlease find the maximal entanglement fidelity $\\max_{\\mathcal{R}} F_e(\\tau,\\mathcal{R}\\circ \\Lambda)$ in terms of an infinite series of $\\gamma$.\nShow your reasoning, then give the final expression as an infinite series in gamma.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "prompt_4", |
| "task_prompt": "Cohomology for Lie Algebra\n\nContext\n\nLet G be a Lie algebra and V a vector space (viewed as a G-module). We aim to compute the third cohomology group H^3(G, V).\n\nOur present goal is to understand what H^3(G, V) looks like and how it can be evaluated or interpreted. By definition, using the Chevalley–Eilenberg complex (C^*(G, V), d), we have the cochain complex\n\n⋯ \\\\xrightarrow{d^{-1}} C^0(G, V) \\\\xrightarrow{d^0} C^1(G, V) \\\\xrightarrow{d^1} C^2(G, V) \\\\xrightarrow{d^2} C^3(G, V) \\\\xrightarrow{d^3} C^4(G, V) ⟶ ⋯\n\nwhere, for each n, the coboundary map d^n : C^n(G, V) → C^{n+1}(G, V) is defined as follows:\n\nGiven c ∈ C^n(G, V), then d^n c ∈ C^{n+1}(G, V) is given by\n\nd^n c(x_1, …, x_{n+1}) = \\\\sum_{1 ≤ i < j ≤ n+1} (-1)^{i+j} c\\\\left( [x_i, x_j], x_1, …, \\\\hat{x}i, …, \\\\hat{x}j, …, x{n+1} \\\\right) - \\\\sum{i=1}^{n+1} (-1)^i x_i · c(x_1, …, \\\\hat{x}i, …, x{n+1})\n\nfor all x_1, …, x_{n+1} ∈ G. Here x_i · c(x_1, …, \\\\hat{x}i, …, x{n+1}) denotes the action of x_i ∈ G on the value c(x_1, …, \\\\hat{x}i, …, x{n+1}) ∈ V according to the G-module structure of V.\n\nBy definition, the third cohomology group is\n\nH^3(G, V) = Z^3(G, V) / B^3(G, V) = \\\\ker(d^3 : C^3(G, V) → C^4(G, V)) / \\\\im(d^2 : C^2(G, V) → C^3(G, V)).\n\nSince neither \\\\ker(d^3 : C^3(G, V) → C^4(G, V)) nor \\\\im(d^2 : C^2(G, V) → C^3(G, V)) is known explicitly, we cannot compute H^3(G, V) directly in the same elementary way.\n\nTherefore, the task is to explicitly (with mathematical details) evaluate H^3(G, V).", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "s0NJgi", |
| "task_prompt": "Take $S$ to be the set of closed tiles in a Cairo pentagonal tiling.\nGiven a subset $T \\subseteq S$, define $A(T)$ to be the set of tiles which meet some tile in $T$ (in at least one vertex).\nTake a vertex $v$ where 4 tiles meet.\nTake $T_0$ to be a set of two tiles which contain $v$ and share an edge, and $T_{i + 1} = A(T_i)$ for all $i$.\nDefine $n_i$ as the size of $T_i$, giving a sequence $(n_i)_i$ with $n_0 = 2$ and $n_1 = 11$.\nWhat are the next 5 values of $n_i$?\nGive your final answer in the form: \"(a,b,c,d,e)\".", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "sdCSCs", |
| "task_prompt": "Some clinical MRI coils have a built-in Hall effect sensor. This is a critical component for their operation. What is its primary function?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_1", |
| "task_prompt": "methyl 4-azidonaphthalene-2-carboxylate was reacted with ethylaminoethanol under 427 nm (LED lamp) in MeCN at rt for 20 h, then concentrated and then dioxane, DBU and N-bromocaprolactam were added and heated at 80 °C to form Intermediate 1. Intermediate 1 was reacted with 1 M NaOH aqueous solution in MeOH to form intermediate 2. Intermediate 2 (3 equiv.) is reacted with 5-(1,3-dioxoisoindolin-2-yl) 1-methyl (tert-butoxycarbonyl)-L-glutamate (1 equiv.) in the presence of (Ir[dF(CF3)ppy]2(dtbpy))PF6 (CAS: 870987-63-6) (1 mol%), Ni(DME)Cl2 (5 mol%), Terpy (5 mol), Ph3P (2.0 equiv.), KHCO3 (0.4 equiv.), acetone/MeCN (2 mL, v/v = 1:1), blue LEDs, ambient temperature, 12 h to form the final product. What is the IUPAC name of the final product?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_10", |
| "task_prompt": "Reaction of 1-methyl-2-{3-phenylbicyclo[1.1.0]butane-1-carbonyl}-1H-imidazole with ethenylbenzene in the presence of Ir(ppy)3 (1 mol%) under blue light 440 nm forms compound 1 as the major product, while the same starting material reacted in the presence of Ir(ppy)3 (1 mol%) and ZnI2 (10 mol%) under blue light 440 nm forms compound 2 as the major product. What will be the structure of compound 2 in SMILES format?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_11", |
| "task_prompt": "Reaction of phthalazine with tert-butyl 1,2-dihydroazete-1-carboxylate in the presence of the bidentate Lewis acid (BDLA) 5,10-dimethyl-5,10-dihydroboranthrene (5 mol%) promotes inverse electron-demand Diels–Alder (IEDDA) reactions to form the final compound. During optimisation, it was found that when tert-butyl 1,2-dihydroazete-1-carboxylate (3 equiv) was added directly and heated at 80 °C under an inert environment overnight, Compound 1 was formed and HPLC showed a total of 6 isomers. When tert-butyl 1,2-dihydroazete-1-carboxylate (1.2 equiv) was instead added slowly over 20 h and heated at 110 °C under an inert environment for 22 h, only one product, Compound 2, was formed. What is Compound 2 in InChI string format?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_2", |
| "task_prompt": "Ethyl 3-(2-formylphenyl)-2-(naphthalen-2-ylmethyl)acrylate was reacted with p-TsNHNH2 (1 eq) in the presence of [Co(TPP)] (0.05 eq), LiOtBu (1.2 eq) and benzene under inert environment at 60 °C overnight, undergoing in-situ generation of a cobalt(III)-carbene radical intermediate and forming final product compound 1. What is the IUPAC name of compound 1?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_4", |
| "task_prompt": "Reaction of estrone with 70% HNO3 in acetic acid forms the p-quinol derivative intermediate 1. Intermediate 1 was reacted with 5 mol% of p-TsOH·H2O in DCE at rt and undergoes a quinol–enedione rearrangement to form the final product. What is the IUPAC name of the final product (do not consider the stereocentre)?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_5", |
| "task_prompt": "ethyl (1S,2R)-1-ethynyl-2-hydroxy-3-methylidene(2-²H)cyclohexane-1-carboxylate was reacted with [RhCl(CO)2]2 (5 mol%) at CO 1 atmosphere at room temperature in anhydrous ClCH2CH2Cl (0.025 M) at 85 °C. The reaction mixture was then stirred at 85 °C for 8 hours, undergoing Rh-catalysed [3+2] cycloaddition to form the final molecule. What is the IUPAC name of the final molecule?. Reference image: refs/task_5_reference_image.png", |
| "reference_file": "task_5_reference_image.png" |
| }, |
| { |
| "task_id": "task_58", |
| "task_prompt": "refs/xray_image_58.png is a chest X-ray image from a curated set of anonymized studies that include both healthy cases and pneumothorax cases with expert segmentation masks. Examine the image and determine whether pneumothorax is present. Your final answer must only contain 'Yes' or 'No' and nothing else.", |
| "reference_file": "xray_image_58.png" |
| }, |
| { |
| "task_id": "task_6", |
| "task_prompt": "1,2,3-cyclohexatriene and substituted derivatives can engage in several types of cycloaddition and nucleophilic addition reactions, providing an unconventional means to access structurally complex molecules. Researchers have made an azacyclic 1,2,3-triene precursor, 1-methyl-6-oxo-4-(triethylsilyl)-1,6-dihydropyridin-3-yl 1,1,2,2,3,3,4,4,4-nonafluorobutane-1-sulfonate, which was reacted with 1,1-diethoxyethene in the presence of 18-crown-6 (2.0 equiv) and KF (2.0 equiv) in an inert environment at 60 °C for 50 hours to form the final product. What is the IUPAC name of the final product?. Reference image: refs/task_6_reference_image.png", |
| "reference_file": "task_6_reference_image.png" |
| }, |
| { |
| "task_id": "task_74", |
| "task_prompt": "refs/xray_image_74.png is a chest X-ray image from a curated set of anonymized studies that include both healthy cases and pneumothorax cases with expert segmentation masks. Examine the image and determine whether pneumothorax is present. Your final answer must only contain 'Yes' or 'No' and nothing else.", |
| "reference_file": "xray_image_74.png" |
| }, |
| { |
| "task_id": "task_75", |
| "task_prompt": "refs/xray_image_75.png is a chest X-ray image from a curated set of anonymized studies that include both healthy cases and pneumothorax cases with expert segmentation masks. Examine the image and determine whether pneumothorax is present. Your final answer must only contain 'Yes' or 'No' and nothing else.", |
| "reference_file": "xray_image_75.png" |
| }, |
| { |
| "task_id": "task_8", |
| "task_prompt": "Deoxycytidine was treated with glyoxal (0.3 equiv.), BF3·OEt2 (0.3 equiv.), sodium nitrite (3.0 equiv.) in dimethylsulfoxide–H2O (9:1) at 37 °C for 12 h to form intermediate 1. Intermediate 1 was treated with 50% N2H4 for 6 h, giving the final product. What is the IUPAC name of the product (do not consider the stereochemistry)?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_9", |
| "task_prompt": "Adenosine was treated with glyoxal (0.3 equiv.), BF3·OEt2 (0.3 equiv.), sodium nitrite (3.0 equiv.) in dimethylsulfoxide–H2O (9:1) at 37 °C for 12 h to form inosine, while when guanosine was reacted under the same conditions it forms Compound 1 instead of xanthosine. To form xanthosine from guanosine requires furfural (1 equiv), BF3·OEt2 (0.3 equiv.), sodium nitrite (3.0 equiv.) in dimethylsulfoxide–H2O (9:1) at 50 °C for 12 h. What is the IUPAC name of Compound 1 (do not consider the stereochemistry)?", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "task_93", |
| "task_prompt": "refs/xray_image_93.png is a chest X-ray image from a curated set of anonymized studies that include both healthy cases and pneumothorax cases with expert segmentation masks. Examine the image and determine whether pneumothorax is present. Your final answer must only contain 'Yes' or 'No' and nothing else.", |
| "reference_file": "xray_image_93.png" |
| }, |
| { |
| "task_id": "task_97", |
| "task_prompt": "refs/xray_image_97.png is a chest X-ray image from a curated set of anonymized studies that include both healthy cases and pneumothorax cases with expert segmentation masks. Examine the image and determine whether pneumothorax is present. Your final answer must only contain 'Yes' or 'No' and nothing else.", |
| "reference_file": "xray_image_97.png" |
| }, |
| { |
| "task_id": "task_98", |
| "task_prompt": "refs/xray_image_98.png is a chest X-ray image from a curated set of anonymized studies that include both healthy cases and pneumothorax cases with expert segmentation masks. Examine the image and determine whether pneumothorax is present. Your final answer must only contain 'Yes' or 'No' and nothing else.", |
| "reference_file": "xray_image_98.png" |
| }, |
| { |
| "task_id": "weDhdu", |
| "task_prompt": "You are an industrial biochemist investigating the application of peptidic natural products derived from hydrolysed sea cucumber collagen as antibiotics. Following functional metabolomic analysis of the extracted peptidome, you have discovered and subsequently isolated a peptide with anti-bacterial activity against pathogenic E. coli.\nA high resolution monoprotonated molecular ion [M+H]+ is observed at m/z = 2428.28596. CID fragmentation of this ion affords daughter ions at m/z = 72.04444, 116.07065, 143.08155, 240.13431, 263.13906, 337.18708, 394.17955, 434.23984, 491.2613, 522.27451, 588.31407, 659.33342, 685.36683, 815.43453, 871.44614, 968.49891, 971.53564, 1065.55167, 1068.58841, 1165.64117, 1166.59935, 1262.69393, 1263.65211, 1360.70488, 1363.74161, 1457.75764, 1460.79438, 1557.84714, 1613.85875, 1743.92645, 1769.95986, 1840.97922, 1907.01877, 1938.03198, 1995.05344, 2035.11373, 2092.10621, 2166.15422, 2189.15897, 2286.21173, 2313.22263, 2357.24885, 2410.2754, 2428.28596. The measurements were acquired on a Bruker Solarix FT-ICR-MS.\nProvide a sequence for the isolated peptide using standard N-to-C convention.", |
| "reference_file": "" |
| }, |
| { |
| "task_id": "zkyjEZ", |
| "task_prompt": "Let $G$ be the subgroup of $Homeo(\\mathbb{R})$ consisting of elements $f$ with the following properties:\n1. $f$ is affine except at a discrete set of breakpoints, which belong to $\\mathbb{Z}[\\frac{1}{2}]$;\n2. the slopes of $f$ outside the breakpoints belong to $2^{\\mathbb{Z}}$;\n3. $f$ commutes with the translation $x \\mapsto x+2$.\n4. elements have to leave $\\mathbb{Z}[\\frac{1}{2}]$ invariant.\n\nLet $a : x \\mapsto x + \\frac{3}{32}$ and $b : x \\mapsto x + \\frac{13}{32}$ be elements of $G$. Consider the free product $G*G$ and let $i, j : G \\to G*G$ be the inclusions into the two factors. Compute the stable commutator length of the element $i(a)j(b)$ in $G*G$. Give your final answer as a fraction in the form: \"a/b\".", |
| "reference_file": "" |
| } |
| ] |
| } |
|
|
