A Quantitative Version of the Gibbard-Satterthwaite theorem for Three Alternatives
... P to weak dependence on irrelevant alternatives: We show that if ni=1 Mi (F ) is small, then in some sense, the question whether the output of F is alternative A or alternative B depends only a little on alternatives other than A and B. Specifically, the probability of changing the outcome of F from ...
... P to weak dependence on irrelevant alternatives: We show that if ni=1 Mi (F ) is small, then in some sense, the question whether the output of F is alternative A or alternative B depends only a little on alternatives other than A and B. Specifically, the probability of changing the outcome of F from ...
Prime-perfect numbers - Dartmouth Math Home
... We note that this lower bound, though of the shape xo(1) , exceeds any fixed power of log x for x sufficiently large. Our upper-bound proof covers a class of numbers somewhat wider than that of the primeperfects. Let rad(n) denote the largest squarefree divisor of n, so that n is prime-perfect if an ...
... We note that this lower bound, though of the shape xo(1) , exceeds any fixed power of log x for x sufficiently large. Our upper-bound proof covers a class of numbers somewhat wider than that of the primeperfects. Let rad(n) denote the largest squarefree divisor of n, so that n is prime-perfect if an ...
Full-Text PDF - EMS Publishing House
... 1. m is a sum of four squares. 2. (3) holds for all A < m: this follows from the induction assumption by switching the roles of m and A. 3. (3) holds for all A ≥ m: this is Euler’s part of the proof. Ad 1. Assume that the theorem holds for all natural numbers < m. If m is not squarefree, say m = m 1 ...
... 1. m is a sum of four squares. 2. (3) holds for all A < m: this follows from the induction assumption by switching the roles of m and A. 3. (3) holds for all A ≥ m: this is Euler’s part of the proof. Ad 1. Assume that the theorem holds for all natural numbers < m. If m is not squarefree, say m = m 1 ...
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a ""bell curve"").The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.