Multiplicative combinatorial properties of return time sets in minimal dynamical systems

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.Comment: 32 page

On the priority vector associated with a fuzzy preference relation and a multiplicative preference relation.

We propose two straightforward methods for deriving the priority vector associated with a fuzzy preference relation. Then, using transformations between multiplicative preference relations and fuzzy preference relations, we study the relationships between the priority vectors associated with these two types of preference relations.pairwise comparison matrix; fuzzy preference relation; priority vector

Bootstrapping vs. Asymptotic Theory in Property and Casualty Loss Reserving

One of the key functions of a property and casualty (P&C) insurance company is loss reserving, which calculates how much money the company should retain in order to pay out future claims. Most P&C insurance companies use non-stochastic (non-random) methods to estimate these future liabilities. However, future loss data can also be projected using generalized linear models (GLMs) and stochastic simulation. Two simulation methods that will be the focus of this project are: bootstrapping methodology, which resamples the original loss data (creating pseudo-data in the process) and fits the GLM parameters based on the new data to estimate the sampling distribution of the reserve estimates; and asymptotic theory, which resamples only the GLM parameters (fitted from an original set of data) from a multivariate normal distribution to estimate the sampling distribution of the reserve estimates. Using Excel, R, and SAS software, the copulas of the GLM parameter estimates from the stochastic methods will be compared to the copula from a multivariate normal distribution. Ultimately, the Value at Risk (VaR) and Tail Value at Risk (TVaR) results from each method’s sampling distribution will be compared to each other, with the goal of showing that the two methods produce significantly different reserve estimates and risk capital estimates at the low end of the reserve distribution. This would answer the question as to whether the asymptotic theory procedure sufficiently approximates real-world scenarios