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One can obtain an addition chain for from an addition chain for by including one additional sum = +, from which follows the inequality () + on the lengths of the chains for and . However, this is not always an equality, as in some cases 2 n {\displaystyle 2n} may have a shorter chain than the one obtained in this way.
The optimal algorithm choice depends on the context (such as the relative cost of the multiplication and the number of times a given exponent is re-used). [2] The problem of finding the shortest addition chain cannot be solved by dynamic programming, because it does not satisfy the assumption of optimal substructure. That is, it is not ...
The smallest n for which an addition-subtraction chain is shorter than the minimal addition chain is n=31, which can be computed in only 6 additions (rather than 7 for the minimal addition chain): a 0 = 1 , a 1 = 2 = 1 + 1 , a 2 = 4 = 2 + 2 , a 3 = 8 = 4 + 4 , a 4 = 16 = 8 + 8 , a 5 = 32 = 16 + 16 , a 6 = 31 = 32 − 1. {\displaystyle a_{0}=1 ...
Let X 1 be dosage "level" and X 2 be the blocking factor furnace run. Then the experiment can be described as follows: k = 2 factors (1 primary factor X 1 and 1 blocking factor X 2) L 1 = 4 levels of factor X 1 L 2 = 3 levels of factor X 2 n = 1 replication per cell N = L 1 * L 2 = 4 * 3 = 12 runs. Before randomization, the design trials look like:
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function , then the characteristic function is the Fourier transform (with sign reversal) of the probability density function.
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function.The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian).
Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observing X(t) at intervals of δ units of time. The random variables X (0), X (δ), X (2δ), ... give the sequence of states visited by the δ-skeleton.