Random Number Generation
... with longer periods are needed (So that cycles can be avoided during the run) It is possible to combine two or more multiplicative congruential generators in such a way that the combined generator has good statistical properties and a longer period ...
... with longer periods are needed (So that cycles can be avoided during the run) It is possible to combine two or more multiplicative congruential generators in such a way that the combined generator has good statistical properties and a longer period ...
Geodesics, volumes and Lehmer`s conjecture Mikhail Belolipetsky
... If P (x) is a cyclotomic polynomial then its Mahler measure is equal to 1. Now Lehmer’s conjecture says that the measures of all other P (x) are separated from 1 by an absolute positive constant which is called Lehmer’s number: Lehmer’s Conjecture. There exists m > 1 such that M (P ) ≥ m for all non ...
... If P (x) is a cyclotomic polynomial then its Mahler measure is equal to 1. Now Lehmer’s conjecture says that the measures of all other P (x) are separated from 1 by an absolute positive constant which is called Lehmer’s number: Lehmer’s Conjecture. There exists m > 1 such that M (P ) ≥ m for all non ...
Full text
... First, we shall look at their origin and find their general expression; then we shall establish some of their properties and give various combinatoric applications. Several results may not have been published previously. The notation of periodic numbers and the notion of arithmetic polynomials will ...
... First, we shall look at their origin and find their general expression; then we shall establish some of their properties and give various combinatoric applications. Several results may not have been published previously. The notation of periodic numbers and the notion of arithmetic polynomials will ...
Full text
... follow, for k = 1,2,..., n. Thus, if the n equations (1) hold for infinitely many n, then A is a palindromic sequence. For the converse, suppose n is a positive integer for which the equations En^k in (2) hold. The equations E„th E„tl + E„t2, EnA + E^2 + En,2> • • • > £w,i + K,2 + m~ + E„t„ readily ...
... follow, for k = 1,2,..., n. Thus, if the n equations (1) hold for infinitely many n, then A is a palindromic sequence. For the converse, suppose n is a positive integer for which the equations En^k in (2) hold. The equations E„th E„tl + E„t2, EnA + E^2 + En,2> • • • > £w,i + K,2 + m~ + E„t„ readily ...
Solving Quadratics
... Let ' s revise our conjecture : The number of U turns is less than or equal to one less than the Degree. There is only a turn when the function changes from inc to dec or dec to inc and indicates a max or a min . ...
... Let ' s revise our conjecture : The number of U turns is less than or equal to one less than the Degree. There is only a turn when the function changes from inc to dec or dec to inc and indicates a max or a min . ...
Factor by Grouping Short-cut
... leading coefficient 10 is used twice with 15 and -8. We will always factor out the factor of x on the left side, so let’s leave that x out of the left side. This would give use the two binomials 10 x 15 and 10 x 8 . If you remove the GCF of each binomial, you are left with the answer. 2x ...
... leading coefficient 10 is used twice with 15 and -8. We will always factor out the factor of x on the left side, so let’s leave that x out of the left side. This would give use the two binomials 10 x 15 and 10 x 8 . If you remove the GCF of each binomial, you are left with the answer. 2x ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.