On fractional probabilistic mean value theorems, fractional counting processes and related results

2015 - 2016The thesis collects the outcomes of the author’s research carried out in the research group Probability Theory and Mathematical Statistics at the Department of Mathematics, University of Salerno, during the doctoral programme “Mathematics, Physics and Applications”. The results are at the interface between Fractional Calculus and Probability Theory. While research in probability and applied fields is now well established and enthusiastically supported, the subject of fractional calculus, i.e. the study of an extension of derivatives and integrals to any arbitrary real or complex order, has achieved widespread popularity only during the past four decades or so, because of its applications in several fields of science, engineering and finance. Moreover, the application of the fractional paradigm to probability theory has been carefully but partially explored over the years, especially from the point of view of stochastic processes. The aim of the thesis is to prove some new theorems at the interface between Mathematical Analysis and Probability Theory, and to study rigorously certain new stochastic processes and statistical models constructed on top of some well-known classical results and then generalized by means of fractional calculus. The dissertation is organized as follows. In Chapter 1 we give an overview about the main ideas that inspire fractional calculus and about the mathematical techniques for dealing with fractional operators and the related special functions and probability distributions. In order to develop certain fractional probabilistic analogues of Taylor’s theorem 1 and mean value theorem, in Chapter 2 we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor’s theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion, we discuss the probabilistic Taylor’s theorem based on fractional Caputo derivatives. In Chapter 3 we consider a fractional counting process with jumps of integer amplitude 1,2,...,k, whose probabilities satisfy a suitable system of fractional differencedifferential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first-passage time of the fractional counting process in an integral form. We then show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability. In Chapter 4 we propose a generalization of the alternating Poisson process from the point of view of fractional calculus. We consider the system of differential equations governing the state probabilities of the alternating Poisson process and replace the ordinary derivative with a fractional one (in the Caputo sense). This produces a fractional 2-state point process, whose probability mass is expressed in terms of the (two-parameter) Mittag-Leffler function. We then show that it can be recovered also by means of renewal theory arguments. We study the limit state probability, and certain proportions involving the fractional moments of the sub-renewal periods of the process. In order to derive new Mittag-Leffler-like distributions related to the considered process, we then exploit a transformation acting on pairs of stochastically ordered random variables, which is an extension of the equilibrium operator and deserves interest in the analysis of alternating stochastic processes. In Chapter 5 we analyse a jump-telegraph process by replacing the classical exponential distribution of the interarrival times which separate consecutive velocity changes (and jumps) with a generalized Mittag-Leffler distribution. Such interarrival times constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based arguments, we obtain the forward and backward transition 2 densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. We conclude the chapter by investigating the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds. Chapter 6 is dedicated to a stochastic model for competing risks involving the MittagLeffler distribution, inspired by fractional random growth phenomena. We prove the independence between the time to failure and the cause of failure, and investigate some properties of the related hazard rates and ageing notions. We also face the general problem of identifying the underlying distribution of latent failure times when their joint distribution is expressed in terms of copulas and the time transformed exponential model. The special case concerning the Mittag-Leffler distribution is approached by means of numerical treatment. We finally adapt the proposed model to the case of a random number of independent competing risks. This leads to certain mixtures of Mittag-Leffler distributions, whose parameters are estimated through the method of moments for fractional moments. [edited by author]La tesi raccoglie i risultati dell’attivit`a di ricerca condotta dall’autore nel gruppo di ricerca Calcolo delle Probabilit`a e Statistica Matematica, presso il Dipartimento di Matematica dell’Universita` di Salerno, nell’ambito del Corso di Dottorato in “Matematica, Fisica e Applicazioni”, XXIX ciclo. I risultati si collocano all’interfaccia tra Calcolo delle Probabilit`a e Calcolo Frazionario. Mentre la ricerca in probabilit`a `e oggi ben consolidata e supportata, il calcolo frazionario, cio`e lo studio della possibilita` di generalizzare il calcolo integrale e il calcolo differenziale classici ad un ordine arbitrario, reale o complesso, ha acquisito notevole popolarita` e importanza nel corso degli ultimi quattro decenni, soprattutto in virtu` delle sue applicazioni in numerosi campi delle scienze e dell’ingegneria. Inoltre, le intersezioni tra calcolo delle probabilita` e calcolo frazionario sono state esplorate con attenzione, ma parzialmente, nel corso degli anni, soprattutto dal punto di vista dei processi stocastici. Lo scopo della tesi `e quello di dimostrare alcuni nuovi teoremi che si collocano all’interfaccia tra l’Analisi Matematica e il Calcolo delle Probabilita`, e di studiare con rigore certi nuovi processi stocastici e modelli statistici costruiti a partire da risultati classici ben noti e poi modificati mediante le tecniche del calcolo frazionario. La tesi `e strutturata come segue. Nel primo capitolo si richiamano alcune nozioni di base e le proprieta` dei principali operatori e delle funzioni del calcolo frazionario, l’integrale di Riemann-Liouville, le derivate di Riemann-Liouville e di Caputo, la funzione di Mittag-Leffler. 1 Nel capitolo 2, al fine di ricavare alcuni analoghi probabilistici di tipo frazionario dei teoremi di Taylor e di Lagrange, `e stata introdotta la distribuzione di equilibrio frazionaria di ordine n definita in termini dell’integrale di Weyl e ne sono state indagate le propriet`a principali. In particolare, si dimostra che la distribuzione di equilibrio frazionaria di ordine n costruita a partire da un’assegnata distribuzione di probabilita`, coincide con questa se e solo se essa `e esponenziale. La distribuzione introdotta viene utilizzata per dimostrare una versione frazionaria dei teoremi di Taylor e del valore medio probabilistici, quest’ultimo applicabile a coppie di variabili aleatorie opportunamente ordinate. Inoltre, si forniscono sia risultati che coinvolgono i momenti normalizzati e un’estensione frazionaria della varianza, sia una formula di interesse nelle scienze attuariali. In conclusione, si discute il teorema di Taylor probabilistico basato sulla derivata frazionaria nel senso di Caputo. Nel terzo capitolo `e stata considerata una generalizzazione frazionaria del processo di Poisson con salti di ampiezza arbitraria, esprimendo la legge di probabilita` mediante funzioni di tipo Mittag- Leffler. L’evoluzione del processo `e guidata da equazioni differenziali e alle differenze finite frazionarie. Dopo aver studiato due rappresentazioni equivalenti del processo considerato, particolare attenzione `e stata posta al problema del tempo di primo passaggio, alla determinazione dei tempi di attesa ed a problemi di tipo asintotico. Tra le altre cose, si `e mostrato che il tempo di prima occorrenza di un salto di ampiezza i, i ∈{1,2,...,k}, k ∈N, `e distribuito come il tempo di prima occorrenza di un evento di un processo di Poisson frazionario di parametro λi > 0, generalizzando, quindi, una importante proprieta` valida nel caso classico. Nel quarto capitolo si propone una generalizzazione del processo di Poisson alternante dal punto di vista del calcolo frazionario, ottenuta sostituendo nel sistema di equazioni differenziali che governa la funzione di probabilita` del processo di Poisson alternante la derivata ordinaria con la derivata frazionaria (nel senso di Caputo) o, equivalentemente, mediante argomenti di teoria del rinnovo. La massa di probabilit`a del nuovo processo `e espressa in termini della funzione di Mittag-Leffler con due parametri. Abbiamo studiato il comportamento asintotico delle probabilit`a di stato e alcune proporzioni che coinvolgono i momenti frazionari dei periodi di rinnovo del processo. Infine, sono state ricavate nuove distribuzioni di tipo Mittag-Leffler relative al processo considerato sfruttando una trasformazione agente su coppie di variabili casuali ordinate stocasticamente, che estende l’operatore equilibrio, di interesse per l’analisi di processi stocastici alternanti. Nel Capitolo 5 si studia un processo stocastico unidimensionale che descrive un moto aleatorio caratterizzato dall’alternarsi di due diverse velocit`a in direzioni opposte. Il processo che regola i cambi di velocit`a (e di direzione) `e il processo di Poisson alternante di tipo frazionario studiato nel capitolo 4. In particolare, nell’istante in cui si verifica un evento di tale processo si compie un salto di ampiezza non aleatoria e quindi il cambiamento di direzione. Pertanto, il processo in esame `e una generalizzazione del processo del telegrafo integrato con salti. Le densit`a di transizione in avanti e all’indietro del moto sono espresse come serie uniformemente convergenti 2 di funzioni di Mittag-Leffler. Particolare attenzione `e stata dedicata al caso di salti di ampiezza costante e uguale distribuzione dei tempi di rinnovo. La distribuzione del tempo di primo passaggio attraverso una barriera costante `e espressa in modo implicito. Tuttavia, in alcuni casi `e data la forma esplicita. L’analisi viene eseguita anche mediante un approccio computazionale. Partendo da fenomeni di crescita di tipo frazionario, nel capitolo 6 abbiamo costruito un modello statistico a rischi competitivi che coinvolge la distribuzione di MittagLeffler. Abbiamo dimostrato l’indipendenza tra il tempo e la causa del fallimento, ed abbiamo indagato alcune propriet`a dei tassi di rischio e delle nozioni di invecchiamento relativi. Abbiamo trattato il problema dell’individuazione della distribuzione sottostante dei tempi di guasto latenti quando la loro distribuzione congiunta `e espressa in termini di copule e mediante il modello TTE (Time Transformed Exponential). Il caso particolare riguardante la distribuzione Mittag-Leffler `e stato trattato numericamente. Il modello proposto `e stato adattato al caso di un numero casuale di rischi in competizione indipendenti. Questo porta ad alcune misture di distribuzioni di tipo Mittag-Leffler, i cui parametri sono stati stimati mediante il metodo dei momenti per momenti frazionari. [a cura dell'autore]XV n.s

TUTTI! - Music Composition as Dialogue

As an engineer, when I could not comprehend a physical phenomenon, I turned to mathematics. As a mathematician, when I could not link sciences to humanity, I turned to music. As a music composer, I no longer see things, I see others. The novel method of music composition presented herein is a first comprehensive framework, system and architectonic template relying on the ideologies of Mikhail Bakhtin's dialogism as well as on research in auditory perception and cognition to create music dialogue as a means of including and engaging participants in musical communication. Beyond immediate artistic intent, I strive to compose music that fosters inclusiveness and collaboration as a relational social gesture in hope that it might incite people and society to embrace their differences and collaborate with the 'others' around them. After probing aesthetics, communication studies and sociology, I argue that dialogism reveals itself well-suited to the aims of the current research. With dialogism as a guiding philosophy, the chapters then look at the relationship between music and language, perception as authorship, intertextuality, the interplay of imagination and understanding, means of arousal in music, mimesis, motion in music and rhythmic entrainment. Employing findings from Gestalt psychology, psychoacoustics, auditory scene analysis, cognition and psychology of expectation, the remaining chapters propose a cognitively informed polyphonic music composition method capable of reproducing the different constituents of dialogic communication by creating and organizing melodic, harmonic, rhythmic and structural elements. Music theory and principles of orchestration then move to music composition as examples demonstrate how dialogue scored between voice-parts provides opportunities for performers to interact with each other and, consequently, engage listeners experiencing the collaboration. As dialogue can be identified in various works, I postulate that the presented Dialogical Music Composition Method can also serve as a method of music analysis. This personal method of composition also supplies tools that other musicians can opt to employ when endeavouring to build balanced dialogue in music. If visibility is key to identity, then composing music that potentially enters into dialogue which each and every voice promotes 'humanity' through inclusivity, yielding a united Tutti

Technology 2000, volume 1

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An alternating renewal process describes the buildup of perceptual segregation

For some ambiguous scenes perceptual conflict arises between integration and segregation. Initially, all stimulus features seem integrated. Then abruptly, perhaps after a few seconds, a segregated perceptual organization emerges. For example, segregation of acoustic features into streams may require several seconds. In behavioral experiments, when a subject’s reports of stream segregation are averaged over repeated trials, one obtains a buildup function, a smooth time course for segregation probability. The buildup function has been said to reflect an underlying mechanism of evidence accumulation or adaptation. During long duration stimuli perception may alternate between integration and segregation. We present a statistical model based on an alternating renewal process that generates buildup functions without an accumulative process. In our model, perception alternates during a trial between different groupings, as in perceptual bistability, with random and independent dominance durations sampled from different percept-specific probability distributions. Using this theory, we describe the short-term dynamics of buildup observed on short trials in terms of the long-term statistics of percept durations for the two alternating perceptual organizations. Our statistical-dynamics model describes well the buildup functions and alternations in simulations of pseudo-mechanistic neuronal network models with percept-selective populations competing through mutual inhibition. Even though the competition model can show history dependence through slow adaptation, our statistical switching model, that neglects history, predicts well the buildup function. We propose that accumulation is not a necessary feature to produce buildup. Generally, if alternations between two states exhibit independent durations with stationary statistics then the associated buildup function can be described by the statistical dynamics of an alternating renewal process

Aerospace medicine and biology: A continuing bibliography with indexes (supplement 385)

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Divergence in Architectural Research

ConCave Ph.D. Symposium 2020: Divergence in Architectural Research, March 5-6, 2020, Georgia Institute of Technology, Atlanta, GA.The essays in this volume have come together under the theme “Divergence in Architectural Research” and present a snapshot of Ph.D. research being conducted in over thirty architectural research institutions, representing fourteen countries around the world. These essays also provide a window into the presentations and discussions that took place March 5-6, 2020, during the ConCave Ph.D. Symposium “Divergence in Architectural Research,” under the auspices of the School of Architecture, Georgia Institute of Technology, in Atlanta, Georgia. On a preliminary reading, the essays respond to the call of divergence by doing just that; they present the great diversity of research topics, methodologies, and practices currently found under the umbrella of “architectural research.” They inform inquiry within architectural programs and across disciplinary concentrations, and also point to the ways that the academy, research methodologies, and the design profession are evolving and encroaching upon one another, with the unspoken hope of encouraging new relationships, reconfiguring previous assumptions about the discipline, and interweaving research and practice