| 31. | The lower bound, though, seems very hard.
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| 32. | Several variants of this lower bound have been proved.
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| 33. | Theoretically, exponential lower bounds have been proved for the DPLL family of algorithms.
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| 34. | In particular, David S . Johnson obtained a lower bound by computer experiment:
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| 35. | Only upper and lower bounds have been invented.
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| 36. | Lower bounds for and have not been improved since 1965 and 1972, respectively.
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| 37. | For others investigations have determined upper and lower bounds, but not exact solutions.
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| 38. | The best-known lower bounds are due to K . F . Roth.
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| 39. | When the Japanese economy fell into a period of prolonged Zero Lower Bound Problem.
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| 40. | These are lower bounds; the typical frequency is expected to be much higher.
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