Andrea Aveni

Duke University | DU

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b -adic representation of x has a ma...

We present two classes of abstract prearithmetics, {AM}M≥1 and {BM}M>0. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic R+=(R+,+,×,≤R+), and the second one is projective with respect to the extended real Diophantine arithmetic R¯=(R¯,+,×,≤R¯). In addition, we have that every AM and every BM is a comple...

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$, the sequence of vectors made by the frequencies of all possibile strings of length $k$ in the $b$-adic representation of $x$ has...

We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M \geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic $\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is weakly projective with re...

We provide a new proof of the fact that the only Species Sampling Models where the probability of observing a new value does depend just on the sample size n and the number of clusters h are the Gibbs-type Processes.

We define a class curves from [0,1] to the complex plane that exhibit a fractal behavior and we manage to find a closed formula for the image of all the rationals in [0,1]. In order to derive this formula, we use some properties of the Polygamma and Zeta functions.

We introduce Tail-Free processes, with particular reference to Pòlya Trees. And we develop some closed formulas for the mean posterior density of particular Pòlya Trees. We also provide some graphical illustrations.

We introduce Classical Kernel Density Estimation, we find optimal bandwidth and Kernel wrt a square loss criterion and we implement an algorithm on Python. We introduce the Bayesian framework and we prove two formulas by Lo and we apply them to Exponential and Normal Kernels and we develop a Python algorithm for the computation of the optimal Bayes...

The aim of this paper is to investigate the relationship between continuous and discrete dynamical systems in a unified approach. In order to do so, we briefly present the recent theory of time scales. In the main part, we find a general solution to homogeneous, constant-coefficients equations in both continuous and discrete time; we focus on the c...