Related papers: Random index of codivisibility
An index is a function that given an election outputs a value between 0 and 1, indicating the extent to which this election has a particular feature. We seek indices that capture agreement, diversity, and polarization among voters in…
We examine the extent to which random samplings from the values of a random set, determine the distribution of the random set itself. We also comment on how, given the statistics of the sampling, to detect the distribution. Several methods…
An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.
We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving…
We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…
We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some…
Universal compression of patterns of sequences generated by independently identically distributed (i.i.d.) sources with unknown, possibly large, alphabets is investigated. A pattern is a sequence of indices that contains all consecutive…
Consider a linear regression model with independent and identically normally distributed random errors. Suppose that the parameter of interest is a specified linear combination of the regression parameters. We prove that the usual…
Randomness is a crucial resource for a broad range of important applications, such as Monte Carlo simulation and computation, generative artificial intelligence and cryptography. But what is randomness? A widely accepted definition has…
The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove…
Motivated by giving a meaning to "The probability that a random integer has initial digit d", we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers.…
While studying a characteristic number of manifolds we noticed that the calculation was simply computing a multiple of a multinomial coefficient. We were, at the time, interested in computing the greatest common divisor of these…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
We build up a directed network tracing links from a given integer to its divisors and analyze the properties of the Google matrix of this network. The PageRank vector of this matrix is computed numerically and it is shown that its…
Random Indexing is a simple implementation of Random Projections with a wide range of applications. It can solve a variety of problems with good accuracy without introducing much complexity. Here we use it for identifying the language of…
The asymptotics, as $n\to\infty$, for the expected number of distinct part sizes in a random composition of an integer n is obtained.
A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the…
We consider a general class of super-additive scores measuring the similarity of two independent sequences of $n$ i.i.d. letters from a finite alphabet. Our object of interest is the mean score by letter $l_n$. By the subadditivity $l_n$ is…
A real is called integer-valued random if no integer-valued martingale can win arbitrarily much capital betting against it. A real is low for integer-valued randomness if no integer-valued martingale recursive in A can succeed on an…
We apply a common measure of randomness, the entropy, in the context of iterated functions on a finite set with n elements. For a permutation, it turns out that this entropy is asymptotically (for a growing number of iterations) close to…