Decompositions of powers of quadratic forms

We analyze the Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main objective is to provide detailed information about their rank and border rank. These forms are of significant importance because of the classical decomposition expressing the space of polynomials of a fixed degree as a direct sum of the spaces of harmonic polynomials multiplied by a power of the quadratic form. Using the fact that the spaces of harmonic polynomials are irreducible representations of the special orthogonal group over the field of complex numbers, we show that the apolar ideal of the s-th power of a non-degenerate quadratic form in n variables is generated by the set of harmonic polynomials of degree s+1. We also generalize and improve upon some of the results about real decompositions, provided by B. Reznick in his notes from 1992, focusing on possibly minimal decompositions and providing new ones, both real and complex. We investigate the rank of the second power of a non-degenerate quadratic form in n variables, which is equal to (n^2+n+2)/2 in most cases. We also study the border rank of any power of an arbitrary ternary non-degenerate quadratic form, which we determine explicitly using techniques of apolarity and a specific subscheme contained in its apolar ideal. Based on results about smoothability, we prove that the smoothable rank of the s-th power of such form corresponds exactly to its border rank and to the rank of its middle catalecticant matrix, which is equal to (s+1)(s+2)/2.

Le politiche giudiziarie inglesi degli anni 2000: mutamento di paradigma e rafforzamento dell'incisività politica dei giudici

The Ph.D. dissertation analyses the reasons for which political actors (governments, legislatures and political parties) decide consciously to give away a source of power by increasing the political significance of the courts. It focuses on a single case of particular significance: the passage of the Constitutional Reform Act 2005 in the United Kingdom. This Act has deeply changed the governance and the organization of the English judicial system, has provided a much clearer separation of powers and a stronger independence of the judiciary from the executive and the legislative. What’s more, this strengthening of the judicial independence has been decided in a period in which the political role of the English judges was evidently increasing. I argue that the reform can be interpreted as a «paradigm shift» (Hall 1993), that has changed the way in which the judicial power is considered. The most diffused conceptions in the sub-system of the English judicial policies are shifted, and a new paradigm has become dominant. The new paradigm includes: (i) stronger separation of powers, (ii) collective (as well as individual) conception of the independence of the judiciary, (iii) reduction of the political accountability of the judges, (iv) formalization of the guarantees of judicial independence, (v) principle-driven (instead of pragmatic) approach to the reforms, and (vi) transformation of a non-codified constitution in a codified one. Judicialization through political decisions represent an important, but not fully explored, field of research. The literature, in particular, has focused on factors unable to explain the English case: the competitiveness of the party system (Ramseyer 1994), the political uncertainty at the time of constitutional design (Ginsburg 2003), the cultural divisions within the polity (Hirschl 2004), federal institutions and division of powers (Shapiro 2002). All these contributes link the decision to enhance the political relevance of the judges to some kind of diffusion of political power. In the contemporary England, characterized by a relative high concentration of power in the government, the reasons for such a reform should be located elsewhere. I argue that the Constitutional Reform Act 2005 can be interpreted as a result of three different kinds of reasons: (i) the social and demographical transformations of the English judiciary, which have made inefficient most of the precedent mechanism of governance, (ii) the role played by the judges in the policy process and (iii) the cognitive and normative influences originated from the European context, as a consequence of the membership of the United Kingdom to the European Union and the Council of Europe. My thesis is that only a full analysis of all these three aspects can explain the decision to reform the judicial system and the content of the Constitutional Reform Act 2005. Only the cultural influences come from the European legal complex, above all, can explain the paradigm shift previously described.

L'interpretazione autentica nei suoi profili teorici ed applicativi

La trattazione cerca di delineare i confini teorici ed applicativi dell’istituto dell’interpretazione autentica, nella chiara consapevolezza che dietro tale tematica si celi il più complesso problema di una corretta delimitazione tra attività di legis-latio e attività di legis-executio. Il fenomeno delle leggi interpretative costituisce infatti nodo nevralgico e punto di intersezione di tre ambiti materiali distinti, ossia la teoria dell’interpretazione, la teoria delle fonti del diritto e la dottrina di matrice liberale della separazione dei poteri. All’interno del nostro ordinamento, nell’epoca più recente, si è assistito ad un aumento esponenziale di interventi legislativi interpretativi che, allo stato attuale, sono utilizzati per lo più come strumenti di legislazione ordinaria. Sotto questo profilo, il sempre più frequente ricorso alla fonte interpretativa può essere inquadrato nel più complesso fenomeno della “crisi della legge” i cui tradizionali requisiti di generalità, astrattezza ed irretroattività sono stati progressivamente abbandonati dal legislatore parallelamente con l’affermarsi dello Stato costituzionale. L’abuso dello strumento interpretativo da parte del legislatore, gravemente lesivo delle posizioni giuridiche soggettive, non è stato finora efficacemente contrastato all’interno dell’ordinamento nonostante l’elaborazione da parte della Corte costituzionale di una serie di limiti e requisiti di legittimità dell’esegesi legislativa. In tale prospettiva, diventano quindi di rilevanza fondamentale la ricerca e l’esame di strategie e rimedi, giurisdizionali ed istituzionali, tali da arginare l’“onnipotenza” del legislatore interprete. A seguito dell’analisi svolta, è maturata la consapevolezza delle potenzialità insite nella valorizzazione della giurisprudenza della Corte Edu, maggiormente incline a sanzionare l’abuso delle leggi interpretative.

I "governi del Presidente"

Gli avvenimenti politico-istituzionali degli ultimi anni hanno determinato un vivace dibattito sulla questione dei cc.dd. “governi del Presidente”, con cui si suole far riferimento ad Esecutivi il cui procedimento di formazione e le cui crisi registrano un interventismo del Presidente della Repubblica superiore a quello richiesto e/o consentito dal dettato costituzionale. Obiettivo dell’elaborato è duplice: fornire un tentativo definitorio della categoria “governi del Presidente”, nonché valutare la compatibilità di questa con la forma di governo parlamentare, soprattutto in considerazione dei recenti sviluppi delle dinamiche politico-istituzionali italiane. Sulla base di questo presupposto, nel presente lavoro si procede a una ricostruzione dell’evoluzione della prassi e delle consuetudini costituzionali in merito all’esercizio dei poteri presidenziali nella gestione delle crisi di governo, con particolare riferimento al procedimento di formazione dell’Esecutivo. La tesi si presenta articolata in due parti, corrispondenti a due differenti periodi storici (cc.dd. Prima e Seconda Repubblica), organizzate per capitoli, relativi alle principali fasi di evoluzione del sistema politico attraversate nel corso della storia della Repubblica. Questa scelta si giustifica in ragione del fatto che il modus operandi del Presidente della Repubblica risulta fortemente condizionato dai mutamenti del contesto politico-istituzionale. Nei singoli paragrafi, dedicati ai presidenti, vengono trattati l’elezione del Presidente, le modalità di formazione dei governi nominati, nonché ulteriori temi la cui disamina è stata ritenuta opportuna allo scopo di fornire una contestualizzazione più adeguata dell’analisi svolta (utilizzo del potere di esternazione e di scioglimento delle Camere, gestione delle attività di politica interna e politica estera, principali avvenimenti dei periodi storici considerati, e così via).

La distribuzione del potere tra le Istituzioni Europee e la procedura di co-decisione. Il caso della riforma della Politica Agricola Comune

Questa ricerca analizza le implicazioni derivanti dall’introduzione della procedura di co-decisione come procedura legislativa ordinaria nel processo di riforma della politica agricola comune. La diversa distribuzione dei poteri tra le istituzioni europee modifica gli assetti istituzionali e fornisce al Parlamento il ruolo di colegislatore in materia agricola. La forma assunta dalla nuova politica agricola europea scaturisce dalla configurazione dei poteri di contrattazione che ciascun attore ha mostrato nella sede dei triloghi negoziali. La ricerca tenta di verificare la accuratezza predittiva di diversi modelli di contrattazione legislativa attraverso il confronto e la verifica degli errori di predizione sui risultati finali di alcune questioni salienti della riforma della Politica Agricola Comune e allo stesso tempo, cerca di identificare il peso del potere del Parlamento europeo in veste di co-legislatore nel processo di riforma della PAC post-2013.

Il diritto alla mobilita' fra dimensione europea, costituzioni statali e autonomie territoriali

La tesi si prefigge l'obbiettivo di offrire una ricostruzione logico sistematica della disciplina giuridica che regola i trasporti pubblici locali in ambito regionale, statale e comunitario, affrontando le principali questioni interpretative e di coordinamento che esse pongono. Nella primo capitolo, viene analizzato l'evoluzione storica della normativa nazionale che regola il trasporto pubblico locale, soffermandosi soprattutto sulla riforma del trasporto pubblico locale introdotte dal d.lgs. 422/1997. Particolare attenzione è stata posta agli aspetti di programmazione e finanziamento nonché alle modalità di gestione del trasporto pubblico locale, in quanto il quadro normativo applicabile è caratterizzato da un’estrema complessità dovuta ai numerosi interventi legislativi. Nel secondo capitolo viene esaminato l'evoluzione dell'intervento comunitario in materi di trasporto pubblico locale, partendo dal (CE) n. 1191/69 che si limitava a disciplinare gli aiuti di Stato, fino alla normativa quadro per il settore (Regolamento (CE) n. 1370/2007). L'obbiettivo è quello di verificare se le scelte del legislatore italiano, per quanto concerne le modalità di gestione del trasporto pubblico locale possano dirsi coerenti con le scelte a livello comunitario previste dal Regolamento (CE) n. 1370/2007. Viene inoltre affronta la questione dell'articolazione della potestà normativa e amministrativa del settore dei trasporti pubblici locali nelle disposizioni del Titolo V della Costituzione. Lo studio si sofferma soprattutto sulla giurisprudenza della Corte costituzionale per tracciare una chiara individuazione del riparto delle competenze tra Stato e Regioni in materia. Infine nell'ultima parte, esamina le diverse problematiche interpretative e applicative della normativa che disciplina il settore del TPL, dovute all'azzeramento della normativa generale dei servizi pubblici locali di rilevanza economica in seguito al referendum abrogativo del 12 e 13 giugno 2011, nonché della illegittimità costituzionale della normativa contenuta nell'art 4 del d.l. n. 138/2011, ad opera della sentenza della Corte costituzionale n. 199/2012.

Il principio di prevedibilità in materia penale. Riflessi sulle garanzie fondamentali di una analisi comparata sulla "cultura del precedente"

Nell’epoca di profonda transizione dell’era «pos-moderna», il modello tradizionale di legalità che conferisce al Parlamento il monopolio in materia penale entra in crisi in ragione della frammentazione delle fonti del diritto - a livello nazionale e sovranazionale - nonché delle spinte di globalizzazione capaci di incidere, ormai, ben oltre il mero piano economico. A fronte della incapacità del legislatore di rispondere in termini effettivi alla urgenza di razionalizzare e riformare il sistema penale, quindi, il potere giudiziario ha assunto una funzione di sostanziale supplenza, generando inevitabili attriti con il nullum crimen e i suoi corollari. Ne deriva che il Giudice di legittimità sempre più si preoccupa di assicurare la prevedibilità dei mutamenti interpretativi, specie se in malam partem, tenendo a mente gli ultimi approdi della giurisprudenza di Strasburgo in riferimento all’art. 7 CEDU. In particolare, tanto nelle motivazioni delle decisioni quanto nei contributi dottrinali, si diffonde graduale il riferimento a istituti propri del modello di common law: la recente riforma dell’art. 618, co 1 bis, c.p.p., del resto, è stata descritta a più voci come l’avvio della diffusione nell’ordinamento interno della c.d. “cultura del precedente”. In tale frangente, diviene utile approfondire l’opportunità dell’accostamento tra il modello di common law e di civil law, conducendo uno studio comparato tra i due ordinamenti quanto all’esercizio del potere punitivo che intende esaminare l’opportunità di acquisire rimedi propri della esperienza secolare del diritto judge-made in un ordinamento governato dalla soggezione del giudice alla legge.

La codificazione del danno ambientale. La sfida del diritto privato tra interessi individuali e interessi collettivi

La tesi ha ad oggetto la tutela dell’ambiente e il ruolo svolto dal diritto privato. In particolare, si affrontano gli aspetti che caratterizzano il danno all’ambiente e, ancor più nel dettaglio, si fa emergere la dimensione plurioffensiva. Si risolvono alcuni nodi e alcune problematiche che si manifestano in relazione alla legittimazione ad agire per danno all’ambiente e, di conseguenza, si focalizza l’attenzione sull’individuazione delle situazioni giuridiche soggettive ad esso correlate: termini come diritto soggettivo, interessi collettivi, interessi diffusi, legittimazione soggettiva e legittimazione oggettiva costituiscono il leitmotiv dell’indagine. Al fine di poter inquadrare sistematicamente il tema ambientale nella sfera del diritto privato, si analizzano i fondamenti del costituzionalismo e del diritto privato moderno, le ragioni dell’esistenza della dicotomia pubblico-privato e della separazione di interessi ad essa corrispondente. Quest’indagine, contenuta nel primo capitolo della tesi, è necessaria per cogliere le motivazioni della tendenziale estromissione (soprattutto nell’ambito delle categorie civilistiche) degli interessi superindividuali, degli interessi collettivi e dei corpi intermedi che ne sono portatori. Nel secondo capitolo si ripercorre, nell’intervallo temporale che va dagli anni ’70 sino alla fine degli anni ’90, l’evoluzione normativa e le principali teorie relative al danno all’ambiente. L’analisi di questi stessi temi, nel periodo intercorrente tra l’inizio del nuovo Millennio e i giorni nostri, è oggetto di indagine anche del terzo ed ultimo capitolo, in cui l’attenzione si sposta inevitabilmente sulla legislazione europea e sulla tutela multilivello dei diritti. Nelle conclusioni del lavoro, infine, ci si interroga sulla funzione giudiziaria e sulla separazione dei poteri nella gestione del problema ambientale, nonché sulla valenza attuale dei diritti soggettivi e dei diritti fondamentali dell’uomo nel costituzionalismo democratico. In questa sede, si illustra e si propone una dimensione istituzionale del diritto soggettivo, declinato in termini di partecipazione democratica.

Statistical properties of Radio Halos and the re-acceleration model

Galaxy clusters occupy a special position in the cosmic hierarchy as they are the largest bound structures in the Universe. There is now general agreement on a hierarchical picture for the formation of cosmic structures, in which galaxy clusters are supposed to form by accretion of matter and merging between smaller units. During merger events, shocks are driven by the gravity of the dark matter in the diffuse barionic component, which is heated up to the observed temperature. Radio and hard-X ray observations have discovered non-thermal components mixed with the thermal Intra Cluster Medium (ICM) and this is of great importance as it calls for a “revision” of the physics of the ICM. The bulk of present information comes from the radio observations which discovered an increasing number of Mpcsized emissions from the ICM, Radio Halos (at the cluster center) and Radio Relics (at the cluster periphery). These sources are due to synchrotron emission from ultra relativistic electrons diffusing through µG turbulent magnetic fields. Radio Halos are the most spectacular evidence of non-thermal components in the ICM and understanding the origin and evolution of these sources represents one of the most challenging goal of the theory of the ICM. Cluster mergers are the most energetic events in the Universe and a fraction of the energy dissipated during these mergers could be channelled into the amplification of the magnetic fields and into the acceleration of high energy particles via shocks and turbulence driven by these mergers. Present observations of Radio Halos (and possibly of hard X-rays) can be best interpreted in terms of the reacceleration scenario in which MHD turbulence injected during these cluster mergers re-accelerates high energy particles in the ICM. The physics involved in this scenario is very complex and model details are difficult to test, however this model clearly predicts some simple properties of Radio Halos (and resulting IC emission in the hard X-ray band) which are almost independent of the details of the adopted physics. In particular in the re-acceleration scenario MHD turbulence is injected and dissipated during cluster mergers and thus Radio Halos (and also the resulting hard X-ray IC emission) should be transient phenomena (with a typical lifetime <» 1 Gyr) associated with dynamically disturbed clusters. The physics of the re-acceleration scenario should produce an unavoidable cut-off in the spectrum of the re-accelerated electrons, which is due to the balance between turbulent acceleration and radiative losses. The energy at which this cut-off occurs, and thus the maximum frequency at which synchrotron radiation is produced, depends essentially on the efficiency of the acceleration mechanism so that observations at high frequencies are expected to catch only the most efficient phenomena while, in principle, low frequency radio surveys may found these phenomena much common in the Universe. These basic properties should leave an important imprint in the statistical properties of Radio Halos (and of non-thermal phenomena in general) which, however, have not been addressed yet by present modellings. The main focus of this PhD thesis is to calculate, for the first time, the expected statistics of Radio Halos in the context of the re-acceleration scenario. In particular, we shall address the following main questions: • Is it possible to model “self-consistently” the evolution of these sources together with that of the parent clusters? • How the occurrence of Radio Halos is expected to change with cluster mass and to evolve with redshift? How the efficiency to catch Radio Halos in galaxy clusters changes with the observing radio frequency? • How many Radio Halos are expected to form in the Universe? At which redshift is expected the bulk of these sources? • Is it possible to reproduce in the re-acceleration scenario the observed occurrence and number of Radio Halos in the Universe and the observed correlations between thermal and non-thermal properties of galaxy clusters? • Is it possible to constrain the magnetic field intensity and profile in galaxy clusters and the energetic of turbulence in the ICM from the comparison between model expectations and observations? Several astrophysical ingredients are necessary to model the evolution and statistical properties of Radio Halos in the context of re-acceleration model and to address the points given above. For these reason we deserve some space in this PhD thesis to review the important aspects of the physics of the ICM which are of interest to catch our goals. In Chapt. 1 we discuss the physics of galaxy clusters, and in particular, the clusters formation process; in Chapt. 2 we review the main observational properties of non-thermal components in the ICM; and in Chapt. 3 we focus on the physics of magnetic field and of particle acceleration in galaxy clusters. As a relevant application, the theory of Alfv´enic particle acceleration is applied in Chapt. 4 where we report the most important results from calculations we have done in the framework of the re-acceleration scenario. In this Chapter we show that a fraction of the energy of fluid turbulence driven in the ICM by the cluster mergers can be channelled into the injection of Alfv´en waves at small scales and that these waves can efficiently re-accelerate particles and trigger Radio Halos and hard X-ray emission. The main part of this PhD work, the calculation of the statistical properties of Radio Halos and non-thermal phenomena as expected in the context of the re-acceleration model and their comparison with observations, is presented in Chapts.5, 6, 7 and 8. In Chapt.5 we present a first approach to semi-analytical calculations of statistical properties of giant Radio Halos. The main goal of this Chapter is to model cluster formation, the injection of turbulence in the ICM and the resulting particle acceleration process. We adopt the semi–analytic extended Press & Schechter (PS) theory to follow the formation of a large synthetic population of galaxy clusters and assume that during a merger a fraction of the PdV work done by the infalling subclusters in passing through the most massive one is injected in the form of magnetosonic waves. Then the processes of stochastic acceleration of the relativistic electrons by these waves and the properties of the ensuing synchrotron (Radio Halos) and inverse Compton (IC, hard X-ray) emission of merging clusters are computed under the assumption of a constant rms average magnetic field strength in emitting volume. The main finding of these calculations is that giant Radio Halos are naturally expected only in the more massive clusters, and that the expected fraction of clusters with Radio Halos is consistent with the observed one. In Chapt. 6 we extend the previous calculations by including a scaling of the magnetic field strength with cluster mass. The inclusion of this scaling allows us to derive the expected correlations between the synchrotron radio power of Radio Halos and the X-ray properties (T, LX) and mass of the hosting clusters. For the first time, we show that these correlations, calculated in the context of the re-acceleration model, are consistent with the observed ones for typical µG strengths of the average B intensity in massive clusters. The calculations presented in this Chapter allow us to derive the evolution of the probability to form Radio Halos as a function of the cluster mass and redshift. The most relevant finding presented in this Chapter is that the luminosity functions of giant Radio Halos at 1.4 GHz are expected to peak around a radio power » 1024 W/Hz and to flatten (or cut-off) at lower radio powers because of the decrease of the electron re-acceleration efficiency in smaller galaxy clusters. In Chapt. 6 we also derive the expected number counts of Radio Halos and compare them with available observations: we claim that » 100 Radio Halos in the Universe can be observed at 1.4 GHz with deep surveys, while more than 1000 Radio Halos are expected to be discovered in the next future by LOFAR at 150 MHz. This is the first (and so far unique) model expectation for the number counts of Radio Halos at lower frequency and allows to design future radio surveys. Based on the results of Chapt. 6, in Chapt.7 we present a work in progress on a “revision” of the occurrence of Radio Halos. We combine past results from the NVSS radio survey (z » 0.05 − 0.2) with our ongoing GMRT Radio Halos Pointed Observations of 50 X-ray luminous galaxy clusters (at z » 0.2−0.4) and discuss the possibility to test our model expectations with the number counts of Radio Halos at z » 0.05 − 0.4. The most relevant limitation in the calculations presented in Chapt. 5 and 6 is the assumption of an “averaged” size of Radio Halos independently of their radio luminosity and of the mass of the parent clusters. This assumption cannot be released in the context of the PS formalism used to describe the formation process of clusters, while a more detailed analysis of the physics of cluster mergers and of the injection process of turbulence in the ICM would require an approach based on numerical (possible MHD) simulations of a very large volume of the Universe which is however well beyond the aim of this PhD thesis. On the other hand, in Chapt.8 we report our discovery of novel correlations between the size (RH) of Radio Halos and their radio power and between RH and the cluster mass within the Radio Halo region, MH. In particular this last “geometrical” MH − RH correlation allows us to “observationally” overcome the limitation of the “average” size of Radio Halos. Thus in this Chapter, by making use of this “geometrical” correlation and of a simplified form of the re-acceleration model based on the results of Chapt. 5 and 6 we are able to discuss expected correlations between the synchrotron power and the thermal cluster quantities relative to the radio emitting region. This is a new powerful tool of investigation and we show that all the observed correlations (PR − RH, PR − MH, PR − T, PR − LX, . . . ) now become well understood in the context of the re-acceleration model. In addition, we find that observationally the size of Radio Halos scales non-linearly with the virial radius of the parent cluster, and this immediately means that the fraction of the cluster volume which is radio emitting increases with cluster mass and thus that the non-thermal component in clusters is not self-similar.

Wavelet analysis of heart rate variability related to nocturnal frontal lobe epilepsy seizures

Introduction: Nocturnal frontal lobe epilepsy (NFLE) is a distinct syndrome of partial epilepsy whose clinical features comprise a spectrum of paroxysmal motor manifestations of variable duration and complexity, arising from sleep. Cardiovascular changes during NFLE seizures have previously been observed, however the extent of these modifications and their relationship with seizure onset has not been analyzed in detail. Objective: Aim of present study is to evaluate NFLE seizure related changes in heart rate (HR) and in sympathetic/parasympathetic balance through wavelet analysis of HR variability (HRV). Methods: We evaluated the whole night digitally recorded video-polysomnography (VPSG) of 9 patients diagnosed with NFLE with no history of cardiac disorders and normal cardiac examinations. Events with features of NFLE seizures were selected independently by three examiners and included in the study only if a consensus was reached. Heart rate was evaluated by measuring the interval between two consecutive R-waves of QRS complexes (RRi). RRi series were digitally calculated for a period of 20 minutes, including the seizures and resampled at 10 Hz using cubic spline interpolation. A multiresolution analysis was performed (Daubechies-16 form), and the squared level specific amplitude coefficients were summed across appropriate decomposition levels in order to compute total band powers in bands of interest (LF: 0.039062 - 0.156248, HF: 0.156248 - 0.624992). A general linear model was then applied to estimate changes in RRi, LF and HF powers during three different period (Basal) (30 sec, at least 30 sec before seizure onset, during which no movements occurred and autonomic conditions resulted stationary); pre-seizure period (preSP) (10 sec preceding seizure onset) and seizure period (SP) corresponding to the clinical manifestations. For one of the patients (patient 9) three seizures associated with ictal asystole were recorded, hence he was treated separately. Results: Group analysis performed on 8 patients (41 seizures) showed that RRi remained unchanged during the preSP, while a significant tachycardia was observed in the SP. A significant increase in the LF component was instead observed during both the preSP and the SP (p<0.001) while HF component decreased only in the SP (p<0.001). For patient 9 during the preSP and in the first part of SP a significant tachycardia was observed associated with an increased sympathetic activity (increased LF absolute values and LF%). In the second part of the SP a progressive decrease in HR that gradually exceeded basal values occurred before IA. Bradycardia was associated with an increase in parasympathetic activity (increased HF absolute values and HF%) contrasted by a further increase in LF until the occurrence of IA. Conclusions: These data suggest that changes in autonomic balance toward a sympathetic prevalence always preceded clinical seizure onset in NFLE, even when HR changes were not yet evident, confirming that wavelet analysis is a sensitive technique to detect sudden variations of autonomic balance occurring during transient phenomena. Finally we demonstrated that epileptic asystole is associated with a parasympathetic hypertonus counteracted by a marked sympathetic activation.

A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.