Bruno Rizzi and Number Theory

Franco Eugeni remembers Bruno Rizzi: in this brief introduction, I would like to remember an afternoon spent in “Roma Tre” with Bruno, since we were both Ordinary Professors at that University. We passed it doing a dense program of work for the next three years. At 6.00 pm, I left for “Roseto degli Abruzzi”. At six o'clock a.m. of the next morning, I still have the voice in my ears. A phone call from the Headmaster Ciro d'Aniello, who told me "The Professor is dead" In that afternoon, Emilio Ambrisi and I went to Rome; we were stunned and desperate. Not many days later, a conference was held at the University of Teramo; it was the conference that we had prepared for Bruno, to celebrate his 60th birthday, but it was his memory. In the opening conference, Prof. Antonino Giambo, also fraternal friend of Bruno, burst into a tearful cry, which expressed in a moment, the senses of friendship and love that we all had for the missing friend! Starting from this work with Fabrizio Maturo, the study of some problems left open by the works of Eugeni and Rizzi will be investigated.SuntoRicordo di Franco Eugeni su Bruno Rizzi: in questa brevissima introduzione vorrei ricordare un pomeriggio passato a Roma Tre con Bruno, visto che allora eravamo entrambi Professori Ordinari in quella Università. Lo passammo a fare un denso programma di lavoro per i successivi tre anni. Alle 18.00 ripartii per Roseto degli Abruzzi. Alle sei del mattino successivo, ho ancora la voce nelle orecchie, una telefonata del Preside Ciro d’Aniello, che mi diceva “Il Professore è morto” Nel pomeriggio Emilio Ambrisi ed io ci recammo a Roma, eravamo attoniti e disperati. Non molti giorni dopo si tenne presso l’Università di Teramo un Convegno, era il Convegno che avevamo preparato per Bruno, per festeggiare i suoi 60 anni, fu invece il ricordo. Nella conferenza di apertura il prof. Antonino Giambò, anche lui, come noi, fraterno amico di Bruno, scoppiò in un pianto dirotto, che espresse in un attimo, i sensi dell’amicizia e dell’amore, che tutti noi avevamo per l’amico scomparso! A partire da questo lavoro con Fabrizio Maturo verrà approfondito lo studio di alcune problematiche rimaste aperte dai lavori di Eugeni e Rizzi

Non-parametric probability distributions embedded inside of a linear space provided with a quadratic metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but it is a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributio

Aggregate Bound Choices about Random and Nonrandom Goods Studied via a Nonlinear Analysis

In this paper, bound choices are made after summarizing a finite number of alternatives. This means that each choice is always the barycenter of masses distributed over a finite set of alternatives. More than two marginal goods at a time are not handled. This is because a quadratic metric is used. In our models, two marginal goods give rise to a joint good, so aggregate bound choices are shown. The variability of choice for two marginal goods that are the components of a multiple good is studied. The weak axiom of revealed preference is checked and mean quadratic differences connected with multiple goods are proposed. In this paper, many differences from vast majority of current research about choices and preferences appear. First of all, conditions of certainty are viewed to be as an extreme simplification. In fact, in almost all circumstances, and at all times, we all find ourselves in a state of uncertainty. Secondly, the two notions, probability and utility, on which the correct criterion of decision-making depends, are treated inside linear spaces over R having a different dimension in accordance with the pure subjectivistic point of vie

The consumer’s demand functions defined to study contingent consumption plans. Summarized probability distributions: a mathematical application to contingent consumption choices

Given two probability distributions expressing returns on two single risky assets of a portfolio, we innovatively define two consumer’s demand functions connected with two contingent consumption plans. This thing is possible whenever we coherently summarize every probability distribution being chosen by the consumer. Since prevision choices are consumption choices being made by the consumer inside of a metric space, we show that prevision choices can be studied by means of the standard economic model of consumer behavior. Such a model implies that we consider all coherent previsions of a joint distribution. They are decomposed inside of a metric space. Such a space coincides with the consumer’s consumption space. In this paper, we do not consider a joint distribution only. It follows that we innovatively define a stand-alone and double risky asset. Different summary measures of it characterizing consumption choices being made by the consumer can then be studied inside of a linear space over ℝ. We show that it is possible to obtain different summary measures of probability distributions by using two different quadratic metrics. In this paper, our results are based on a particular approach to the origin of the variability of probability distributions. We realize that it is not standardized, but it always depends on the state of information and knowledge of the consumer

Tensors Associated with Mean Quadratic Differences Explaining the Riskiness of Portfolios of Financial Assets

Bound choices such as portfolio choices are studied in an aggregate fashion using an extension of the notion of barycenter of masses. This paper answers the question of whether such an extension is a natural fashion of studying bound choices or not. Given n risky assets, the question of why it is appropriate to treat only two risky assets at a time inside the budget set of the decision-maker is handled in this paper. Two risky assets are two goods. They are two marginal goods. The question of why they always give rise to a joint good inside the budget set of the decision-maker is addressed by this research work. A single risky asset is viewed as a double one using four nonparametric joint distributions of probability. The variability of a joint distribution of probability always depends on the state of information and knowledge associated with a given decision-maker. For this reason, two variability tensors are defined to identify the riskiness of the same risky asset. A multilinear version of the Sharpe ratio is shown. It is based on tensors. After computing the expected return on an n-risky asset portfolio, its riskiness is obtained using mean quadratic differences developed through tensor

Pooling random forest and functional data analysis for biomedical signals supervised classification: theory and application to electrocardiogram data

Scientific progress has contributed to creating many devices to gather vast amounts of biomedical data over time. The goal of these devices is generally to monitor people's health conditions, diagnose, and prevent patients' diseases, for example, to discover cardiovascular disorders or predict epileptic seizures. A common way of investigating these data is classification, but these instruments generate signals often characterized by high dimensionality. Learning from these data is definitely a challenging task due to many issues, for example, the trade-off between complexity and accuracy and the course of dimensionality. This study proposes a supervised classification method based on the joint use of functional data analysis, classification trees, and random forest to deal with massive biomedical data recorded over time. For this purpose, this research suggests different original tools to extract features and train functional classifiers, interpret the classification rules, assess leaves' quality and composition, avoid the classical drawbacks due to the COD, and improve the accuracy of the functional classifiers. Focusing on ECG data as a possible example, the final purpose of this study is to offer an original approach to identify and classify patients at risk using different types of biomedical signals. The results confirm that this line of research is exciting; indeed, the interpretative tools show evidence to be very useful for understanding classification rules. Furthermore, the performance of the proposed functional classifier, in terms of accuracy, is excellent because the latter breaks the previous classification record regarding a well-known ECG dataset

Combining unsupervised and supervised learning techniques for enhancing the performance of functional data classifiers

This paper offers a supervised classification strategy that combines functional data analysis with unsupervised and supervised classification methods. Specifically, a two-steps classification technique for high-dimensional time series treated as functional data is suggested. The first stage is based on extracting additional knowledge from the data using unsupervised classification employing suitable metrics. The second phase applies functional supervised classification of the new patterns learned via appropriate basis representations. The experiments on ECG data and comparison with the classical approaches show the effectiveness of the proposed technique and exciting refinement in terms of accuracy. A simulation study with six scenarios is also offered to demonstrate the efficacy of the suggested strategy. The results reveal that this line of investigation is compelling and worthy of further development

Approach of the value of an annuity when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions

This paper proposes an expression of the value of an annuity with payments of 1 unit each when the interest rate is random. In order to attain this objective, we proceed on the assumption that the non-central moments of the capitalization factor are known. Specifically, to calculate the value of these annuities, we propose two different expressions. First, we suppose that the random interest rate is normally distributed; then, we assume that it follows the beta distribution. A practical application of these two methodologies is also implemented using the R statistical software

L’Item Response Theory come strumento di valutazione delle eccellenze nella scuola

According to the modern human capital theory, the learning process is assuming a central role for a country development. Therefore, it is very important to encourage an active and conscious learning. This paper shows the case of the National Olympic of Statistics, that is an initiative to improve statistic reasoning in the Second Grade Middle School. In particular, it has been analyzed, through Item Response Theory models, only one of the competition questionnaires. The aim of this work is to provide a critical review of the questionnaire and to highlight the importance of using statistical methods in evaluation processes