3 research outputs found
Benford Behavior of a Higher-Dimensional Fragmentation Process
Nature and our world have a bias! Roughly of the time the number
occurs as the leading digit in many datasets base . This phenomenon is
known as Benford's law and it arrises in diverse fields such as the stock
market, optimizing computers, street addresses, Fibonacci numbers, and is often
used to detect possible fraud. Based on previous work, we know that different
forms of a one-dimensional stick fragmentation result in pieces whose lengths
follow Benford's Law. We generalize this result and show that this can be
extended to any finite-dimensional ``volume''. We further conjecture that even
lower-dimensional volumes, under the unrestricted fragmentation process, follow
Benford's Law
Benfordness of the Generalized Gamma Distribution
The generalized gamma distribution shows up in many problems related to
engineering, hydrology as well as survival analysis. Earlier work has been done
that estimated the deviation of the exponential and the Weibull distribution
from Benford's Law. We give a mathematical explanation for the Benfordness of
the generalized gamma distribution and present a measure for the deviation of
the generalized gamma distribution from the Benford distribution
Probabilistic Parking Functions
We consider the notion of classical parking functions by introducing
randomness and a new parking protocol, as inspired by the work presented in the
paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by
Carlson, Christensen, Harris, Jones, and Rodr\'iguez. Among our results, we
prove that the probability of obtaining a parking function, from a length
preference vector, is independent of the probabilistic parameter . We also
explore the properties of a preference vector given that it is a parking
function and discuss the effect of the probabilistic parameter . Of special
interest is when , where we demonstrate a sharp transition in some
parking statistics. We also present several interesting combinatorial
consequences of the parking protocol. In particular, we provide a combinatorial
interpretation for the array described in OEIS A220884 as the expected number
of preference sequences with a particular property related to occupied parking
spots, which solves an open problem of Novelli and Thibon posed in 2020
(arXiv:1209.5959). Lastly, we connect our results to other weighted phenomena
in combinatorics and provide further directions for research.Comment: 22 pages, 3 figures, 4 table