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"value": "Consider the pure state\\n\\n\\[\\n|\\psi\\rangle := \\alpha |00\\rangle + \\beta |11\\rangle\\n\\]\\n\\nin the Hilbert space \\(\\mathbb{C}^2 \\otimes \\mathbb{C}^2\\), where \\(\\alpha, \\beta \\in \\mathbb{C}\\) and \\(|\\alpha|^2 + |\\beta|^2 = 1\\). Let \\(\\rho := |\\psi\\r... |
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"value": "4. Consider \\( y'' + \\lambda y = 0 \\) subject to the periodic boundary conditions \\( y(-L) = y(L), y'(-L) = y'(L) \\). Show that the eigenfunctions are\\n\\n\\[\\left\\{ 1, \\cos \\frac{\\pi x}{L}, \\cos \\frac{2\\pi x}{L}, \\ldots \\sin \\frac{\\pi x}{L}, \\sin \\frac{2\\pi ... |
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"value": "Two students were asked to write an nth term for the sequence 1, 16, 81, 256, ... and to write the 5th term of the sequence. One student gave the nth term as \\(u_n = n^4\\). The other student, who did not recognize this simple law of formation, wrote \\(u_n = 10n^3 - 35n^2 + 50n... |
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"value": "Consider a box containing an ideal classical gas at pressure \\( P \\) and temperature \\( T \\). The walls of the box have \\( N_0 \\) absorbing sites, each of which can absorb one molecule of the gas. Let \\(-\\epsilon\\) be the energy of an absorbed molecule.\\n\\n (a) Find t... |
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"value": "Draw electron dot diagrams for \\( \\text{H}_3\\text{PO}_4 \\), \\( \\text{H}_3\\text{PO}_3 \\), and \\( \\text{H}_3\\text{PO}_2 \\). *Hint: Four atoms are bonded directly to the phosphorus atom in each case.*"
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"value": "<think>\nOkay, so I n... |
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"value": "Prove the Riesz Representation Theorem: Assume \\( H \\) is a Hilbert space with scalar product \\( \\langle \\cdot, \\cdot \\rangle \\). For every continuous linear functional \\( L \\) on \\( H \\) there exists a unique \\( v \\in H \\) such that for all \\( u \\in H \\) \\[ L(... |
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"value": "Prove that \\( \\mathbf{A} \\times (\\mathbf{B} + \\mathbf{C}) = \\mathbf{A} \\times \\mathbf{B} + \\mathbf{A} \\times \\mathbf{C} \\) in the general case where **A**, **B**, and **C** are non-coplanar."
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"value": "<think>\nOkay, so I need to ... |
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"value": "1250 cm^3/s of water is to be pumped through a steel pipe, 25 mm diameter and 30 m long, to a tank 12 m higher than its reservoir. Calculate the approximate power required. What type of pump would you install for the purpose and what power motor (in kW) would you provide? Viscosi... |
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"value": "Show that (0, 0) is a stable critical point of the nonlinear plane autonomous system\\n\\nx' = -y - x\\sqrt{x^2 + y^2}\\ny' = x - y\\sqrt{x^2 + y^2}\\n\\nconsidered in Example 6 of Section 10.1."
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"value": "<think>\nOkay, so I need to show tha... |
[{"from":"user","value":"Aniline and nitrosobenzene, in the presence of a base, react to give azoben(...TRUNCATED) |
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