Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods

In this paper we are concerned with the analysis of a class of geometric integrators, at first devised in [14, 18], which can be regarded as an energy-conserving variant of Gauss collocation methods. With these latter they share the property of conserving quadratic first integrals but, in addition, they also conserve the Hamiltonian function itself. We here reformulate the methods in a more convenient way, and propose a more refined analysis than that given in [18] also providing, as a by-product, a practical procedure for their implementation. A thorough comparison with the original Gauss methods is carried out by means of a few numerical tests solving Hamiltonian and Poisson problems.Comment: 28 pages, 2 figures, 4 table

Protocollo di autenticazione STS e IKE

Lo scambio delle chiavi di Diffie-Hellman fu il primo sistema a permettere a due interlocutori di condividere una chiave segreta senza l'utilizzo di un canale sicuro. La debolezza più grande di questo sistema è rappresentata dall'attacco "Man in the middle", in cui l'avversario riesce a porsi in mezzo alla comunicazione e impersonare i due interlocutori. Il protocollo STS (Station To Station) nasce nel 1987 per garantire maggiore sicurezza al protocollo base di Diffie-Hellman. Esso, grazie all'utilizzo della firma digitale, aggiunge l'autenticazione dei soggetti coinvolti nello scambio, impedendo così l'attacco Man in the middle. Rimangono, però, ancora possibili attacchi come quello di Lowe. Il protocollo STS, inoltre, è una delle basi per il protocollo di autenticazione standard per la sicurezza Internet, chiamato "Internet Key Exchange (IKE)". A questo proposito introduciamo lo standard ISO/OSI, composto da 7 livelli, e il protocollo IP che lavora al livello 3. Quest'ultimo si occupa dell'indirizzamento e instradamento dei dati che viaggiano in rete. IP è stato migliorato in IPSec per essere più resistente agli attacchi. L'autenticazione delle parti in IPSec è dovuta al protocollo IKE (Internet Key Exchange). Esistono diverse varianti di questo protocollo, in questa tesi vedremo soltanto le modalità basate sulla firma digitale

Line Integral Solution of Hamiltonian Systems with Holonomic Constraints

In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustrate theoretical findings.Comment: 30 pages, 3 figures, 4 table

Adaptive Regularization for Nonconvex Optimization Using Inexact Function Values and Randomly Perturbed Derivatives

A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function with Lipschitz continuous $p$-th derivative and given an arbitrary optimality order $q \leq p$, it is shown that this algorithm will, in expectation, compute such a point in at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}\epsilon_j\right)^{-\frac{p+1}{p-q+1}}\right)$ inexact evaluations of $f$ and its derivatives whenever $q\in\{1,2\}$, where $\epsilon_j$ is the tolerance for $j$th order accuracy. This bound becomes at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}\epsilon_j\right)^{-\frac{q(p+1)}{p}}\right)$ inexact evaluations if $q>2$ and all derivatives are Lipschitz continuous. Moreover these bounds are sharp in the order of the accuracy tolerances. An extension to convexly constrained problems is also outlined.Comment: 22 page

Adaptive Regularization Algorithms with Inexact Evaluations for Nonconvex Optimization

A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is $\beta$-H\"{o}lder continuous. It features a very flexible adaptive mechanism for determining the inexactness which is allowed, at each iteration, when computing objective function values and derivatives. The complexity analysis covers arbitrary optimality order and arbitrary degree of available approximate derivatives. It extends results of Cartis, Gould and Toint (2018) on the evaluation complexity to the inexact case: if a $q$th order minimizer is sought using approximations to the first $p$ derivatives, it is proved that a suitable approximate minimizer within $\epsilon$ is computed by the proposed algorithm in at most $O(\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ iterations and at most $O(|\log(\epsilon)|\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ approximate evaluations. An algorithmic variant, although more rigid in practice, can be proved to find such an approximate minimizer in $O(|\log(\epsilon)|+\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ evaluations.While the proposed framework remains so far conceptual for high degrees and orders, it is shown to yield simple and computationally realistic inexact methods when specialized to the unconstrained and bound-constrained first- and second-order cases. The deterministic complexity results are finally extended to the stochastic context, yielding adaptive sample-size rules for subsampling methods typical of machine learning.Comment: 32 page

Phenotype correlation in basal cell naevus syndrome (BCNS): our experience

Tale progetto di ricerca ha lo scopo di confrontare le caratteristiche fenotipiche dermatologiche osservate in 2 gruppi di pazienti affetti da sindrome del nevo basocellualre (BCNS): 12 adulti e 4 bambini. Da tale confronto sono emersi dati molto interessanti nella popolazione pediatrica, come la precocità di esordio degli epiteliomi basocellulari (BCC), particolarmente frequenti nelle forme atipiche, ovvero BCC acrochordon-like e BCC localizzati in sedi acrali, il riscontro di tricoepiteliomi, mai descritti ad oggi come reperto associato alla BCNS e la presenza delle due varianti di pits palmo-plantari, rossi e bruni. Trattandosi di una patologia riconosciuta nell’ambito delle malattie rare, la nostra casistica è ridotta pertanto sono necessari ulteriori studi per capire meglio i meccanismi che sottendono a tali differenze. Tuttavia in letteratura lavori di confronto tra le differenti espressioni della BCNS nelle diverse epoche di vita sono scarsi e pertanto ci è sembrato interessante esporre i nostri risultati. Tale progetto è stato realizzato grazie alla collaborazione multidisciplinare dell’Ambulatorio delle Malattie Rare e del Centro Tumori Cutanei della Dermatologia, della Genetica Medica del Dipartimento di Medicina Specialistica, Diagnostica e Sperimentale (DIMES) e della Chirurgia Orale e Maxillo-Facciale del Dipartimento di Scienze Biomediche e Neuro-Motorie (DIBINEM) dell’Università di Bologna.The purpose of this project of research is to analyse the dermatological phenotypic findings observed in 2 groups of patients affected by basal cell nevus syndrome (BCNS): 12 adults and 4 children. Very interesting data have emerged in the paediatric population, such as the early onset of basal cell carcinomas (BCCs), almost in atypical presentations, namely acronordon-like BCCs and acral- BCCs. In addiction we observed the trichoepitheliomas, never described before associated with the BCNS and the palmar-plantar pits, reds and browns. In the literature studies about the different clinical phenotypes of the BCNS comparing adults and children are rare. Unfortunately, we present a small sample size but BCNS is a very rare disorder. However, further studies are needed to better understand the mechanisms underlying such differences. This project is the result of a multidisciplinary collaboration of the Dermatology Unit (Rare Diseases Center and Cutaneous Tumors Center), Medical Genetics Unit and of Oral and Maxillofacial Surgery Unit of the University of Bologn