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The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. [ 1][ 2][ 3] It is a divide-and-conquer algorithm that reduces the multiplication of two n -digit numbers to three multiplications of n /2-digit numbers and, by repeating this reduction, to at most single-digit ...
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1. An abstract group defined by this multiplication is often denoted C n, and we say that G is isomorphic to the standard cyclic group C n.
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
Thus, a multiply-with-carry generator is a Lehmer generator with modulus p and multiplier b−1 (mod p ). This is the same as a generator with multiplier b, but producing output in reverse order, which does not affect the quality of the resultant pseudorandom numbers.
They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered. Sometimes the notation M 8, M 9, M 10, M 20, and M 21 is used for related groups (which act on sets of 8, 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While ...
Consequently, the time can be determined to 1 part in 10,000. Interpolators are often used with a stable system clock. The start event is asynchronous, but the stop event is a following clock. [9] [11] For convenience, imagine that the fast ramp rises exactly 1 volt during a 100 ns clock period. Assume the start event occurs at 67.3 ns after a ...
Another more complicated belt machine was invented in 1903 by Juan Burboa [1] [5] A more immediate inspiration for Van de Graaff was a generator W. F. G. Swann was developing in the 1920s in which charge was transported to an electrode by falling metal balls, thus returning to the principle of the Kelvin water dropper. [1] [6]