arXiv subjects
Publications and source records attributed to Andrea Berdondini.
This book consists of a selection of articles divided into three main themes: Statistics, Quantitative Trading, Psychology. These three arguments are indispensable for the development of a quantitative trading system. The order of the articles was chosen so as to constitute a single logical reasoning that develops progressively.
ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative numbers. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only arrangement that minimizes the distance between two powers having the same absolute value of the base D, with |D|<=P. This arrangement is considered important because by increasing the number of powers of the prime numbers within a range of consecutive numbers, it is presumed to decrease the overlap between the prime numbers considered. Consequently, by reducing these overlaps, we suppose to obtain an arrangement, in which the prime numbers less than and equal to P and their multiples occupy the greatest possible number of positions within a range of 2P+1 consecutive numbers. If this result could be demonstrated, would imply not only the resolution of the Legendre conjecture, but also a step forward in the resolution of the twin primes conjecture and the Goldbach conjecture.
In this article we will propose a completely new point of view for solving one of the most important paradoxes concerning game theory. The solution develop shifts the focus from the result to the strategy s ability to operate in a cognitive way by exploiting useful information about the system. In order to determine from a mathematical point of view if a strategy is cognitive, we use Von Mises' axiom of randomness. Based on this axiom, the knowledge of useful information consequently generates results that cannot be reproduced randomly. Useful information in this case may be seen as a significant datum for the recipient, for their present or future decision-making process. Finally, by resolving the paradox from this new point of view, we will demonstrate that an expected gain that tends toward infinity is not always a consequence of a cognitive and non-random strategy. Therefore, this result leads us to define a hierarchy of values in decision-making, where the cognitive aspect, whose statistical consequence is a divergence from random behaviour, turns out to be more important than the expected gain.
In this article, I will present a paradox whose purpose is to draw your attention to an important topic in finance, concerning the non-independence of the financial returns (non-ergodic hypothesis). In this paradox, we have two people sitting at a table separated by a black sheet so that they cannot see each other and are playing the following game: the person we call A flip a coin and the person we'll call B tries to guess the outcome of the coin flip. At the end of the game, both people are asked to estimate the compound probability of the result obtained. The two people give two different answers, one estimates the events as independent and the other one considers the events as dependent therefore they calculate the conditional probability differently. This paradox show how the erroneous estimation of conditional probability implies a strong distortion of the forecasting skill that can lead us to bear excessive risks.
In this article I present results from a statistical study of prime numbers that shows a behaviour that is not compatible with the thesis that they are distributed randomly. The analysis is based on studying two arithmetical progressions defined by the following polynomials: ($1+6n$, $5+6n$, $n\in{N}$) whose respective numerical sequences have the characteristic of containing all the prime numbers except $3$ and $2$. If prime numbers were distributed randomly, we would expect the two polynomials to generate the same number of primes. Instead, as the reported findings show, we note that the polynomial $5+6n$ tends to generate many more primes, and that this divergence grows progressively as more prime numbers are considered. A possible explanation for this phenomenon can be found by calculating the number of products that generate composite numbers which are expressible by the two polynomials. This analysis reveals that the number of products that generate composite numbers expressible by the polynomial $1+6n$ is $(n+1)^{ 2}$, while the number of products that generate composites expressible by the polynomial $5+6n$ is $(n+1)n$, con $n\in{N}$. As a composite number is a non-prime number, this difference incited me to analyse the distribution of prime numbers generated by the two polynomials. The results, based on studying the first (approx.) 500 million prime numbers, confirm the fact that the number of primes that can be written using the polynomial $1+6n$ is lower than the number of primes that can written using the polynomial $5+6n$, and that this divergence grows progressively with the number of primes considered.