A well-known, classical conundrum, which is related to conditional probability, has heretofore only been used for games and puzzles. It is shown here, both empirically and formally, that the counterintuitive phenomeno...A well-known, classical conundrum, which is related to conditional probability, has heretofore only been used for games and puzzles. It is shown here, both empirically and formally, that the counterintuitive phenomenon in question has consequences that are far more profound, especially for physics. A simple card game the reader can play at home demonstrates the counterintuitive phenomenon, and shows how it gives rise to hidden variables. These variables are “hidden” in the sense that they belong to the past and no longer exist. A formal proof shows that the results are due to the duration of what can be thought of as a gambler’s bet, without loss of generalization. The bet is over when it is won or lost, analogous to the collapse of a wave function. In the meantime, new and empowering information does not change the original probabilities. A related thought experiment involving a pregnant woman demonstrates that macroscopic systems do not always have states that are completely intrinsic. Rather, the state of a macroscopic system may depend upon how the experiment is set up and how the system is measured even though no wave functions are involved. This obviously mitigates the chasm between the quantum mechanical and the classical.展开更多
In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always e...In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.展开更多
文摘A well-known, classical conundrum, which is related to conditional probability, has heretofore only been used for games and puzzles. It is shown here, both empirically and formally, that the counterintuitive phenomenon in question has consequences that are far more profound, especially for physics. A simple card game the reader can play at home demonstrates the counterintuitive phenomenon, and shows how it gives rise to hidden variables. These variables are “hidden” in the sense that they belong to the past and no longer exist. A formal proof shows that the results are due to the duration of what can be thought of as a gambler’s bet, without loss of generalization. The bet is over when it is won or lost, analogous to the collapse of a wave function. In the meantime, new and empowering information does not change the original probabilities. A related thought experiment involving a pregnant woman demonstrates that macroscopic systems do not always have states that are completely intrinsic. Rather, the state of a macroscopic system may depend upon how the experiment is set up and how the system is measured even though no wave functions are involved. This obviously mitigates the chasm between the quantum mechanical and the classical.
文摘In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.
