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In the second line, the number one is added to the fraction, and again Excel displays only 15 figures. In the third line, one is subtracted from the sum using Excel. Because the sum in the second line has only eleven 1's after the decimal, the difference when 1 is subtracted from this displayed value is three 0's followed by a string of eleven 1's.
If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these.
A confidence interval is a range of values that can contain the true real parameter value in given a certain level of confidence. The first step is to estimate the best-unbiased estimate of the population parameter.
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1.
The neutral value of the pH depends on the temperature and is lower than 7 if the temperature increases above 25 °C. The pH range is commonly given as zero to 14, but a pH value can be less than 0 for very concentrated strong acids or greater than 14 for very concentrated strong bases. [2]
Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346 × 10 −23 J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901. [11]
Word2vec was created, patented, [7] and published in 2013 by a team of researchers led by Mikolov at Google over two papers. [1] [2] The original paper was rejected by reviewers for ICLR conference 2013. It also took months for the code to be approved for open-sourcing. [8] Other researchers helped analyse and explain the algorithm. [4]
The Kaldor–Hicks criterion is widely applied in game theory's non-zero sum games, such as DOTMLPF, welfare economics, and managerial economics. For example, it forms an underlying rationale for cost–benefit analysis. In cost–benefit analysis, a project (for example, a new airport) is evaluated by comparing the total costs, such as ...