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# Doctoral School of Sciences and Innovative Technologies

Curriculum Vitae

### Tesi di dottorato

Title. Approximations of irrationalities by using linear recurrent sequences.

Abstract. The thesis is devoted to the study of quadratic and cubic irrationalities. In this context, Rédei rational functions (which are defined as ratio of recurrent sequences) are originally introduced like convergents of certain periodic continued fractions with rational partial quotients. Among these convergents, Newton and Padé approximations of \sqrt{d}  arise simultaneously, for every positive integer d not square. Rédei rational functions have been successively generalized in order to approximate cubic irrationalities and to approach the Hermite problem. In this way, similar results have been obtained for any cubic roots.

### Attività di ricerca

Publications.

1. E. Bellini, N. Murru, An efficient and secure RSA-like cryptosystem exploiting Rédei rational functions over conics, Finite Fields and their Applications, Vol. 39, 179-194, 2016.
2. N. Murru, R. Rossini, A Bayesian approach for initialization of weights in backpropagation neural net with application to character recognition, Neurocomputing, Vol. 193, 92-105, 2016.
3. G. Airò Farulla, T. Armano, A. Capietto, N. Murru, R. Rossini, Artificial neural networks and fuzzy logic for recognizing alphabet characters and mathematical symbols, Lecture Notes in Computer Science, Vol. 9759, Computers Helping People with Special Needs, 7-14, 2016.
4. M. Abrate, S. Barbero, U. Cerruti, N. Murru, The biharmonic mean, Mathematical Reports, Vol. 18, No. 4, 2016
5. N. Murru, Linear recurrence sequences and periodicity of multidimensional continued fractions, The Ramanujan Journal, Ready Online, 2016.
6. N. Murru, On the periodic writing of cubic irrationals and a generalization of Rédei functions, International Journal of Number Theory, Vol. 11, No. 3, 779-799, 2015.
7. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, Article A38, 2015.
8. S. Chiaradonna, F. Di Giandomenico, N. Murru, On enhancing efficiency and accuracy of particle swarm optimization algorithms, International Journal on Innovative Computing, Information and Control, Vol. 11. No. 4, 1165-1189, 2015.
9. T. Armano, A. Capietto, N. Murru, R. Rossini, E. Tornavacca, Accessibility and inclusiveness of mathematics in school and business training courses, Mondo Digitale, 2015.
10. N. Murru, Periodic representations and rational approximations for quadratic irrationalities by means of Rédei rational functions, Journal of Algebra, Number Theory and Applications, Vol. 33, No. 2, 141-154, 2014.
11. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Colored compositions, Invert operator and elegant compositions with the “black tie", Discrete Mathematics, Vol. 335, 1-7, 2014.
12. C. Di Sarno, N. Murru, A. Garofalo, G. Cerullo, F. Di Giandomenico, S. Chiaradonna, Power Grid Outlier Treatment through Kalman Filter, 2014 IEEE International Symposium on Software Reliability Engineering Workshops,  Naples, Italy, 407 - 412, 3-6 Nov, 2014
13. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Writing pi as sum of arctangents with linear recurrent sequences, Golden mean and Lucas numbers, International Journal of Number Theory, Vol. 10, No. 5, 1309-1319, 2014.
14. S. Chiaradonna, F. Di Giandomenico, N. Murru, On a Modeling Approach to Analyze Resilience of a Smart Grid Infrastructure, Dependable Computing Conference (EDCC), 2014 Tenth European, Newcastle, United Kingdom, 166-177, 13-16 May, 2014.
15. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Identities involving zeros of Ramanujan and Shanks cubic polynomials, Journal of Integer Sequences, Vol. 16, Article 13.8.1, 2013.
16. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Periodic representations and rational approximations of square roots, Journal of Approximation Theory, Vol. 175, 83-90, 2013.
17. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Construction and composition of rooted trees via descent functions, Algebra, Vol. 2013, Article 543913, 2013.
18. G. E. D'Errico, N. Murru, In-process estimation of time-variant contingently correlated measurands, International Journal of Metrology and Quality Engineering, Vol. 3, Issue 03, 2012.
19. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Periodic representations for cubic irrationalities, The Fibonacci Quarterly, Vol. 50, No. 3, 252-264, 2012.
20. S. Barbero, U. Cerruti, N. Murru, Squaring the magic squares of order 4, Journal of Algebra, Number Theory: Advances and Applications, Vol. 7, No. 1, 31-46, 2012.
21. G. E. D'Errico, N. Murru, Real–time estimation of dynamic multi-dimensional measurands, Proccedings of 20th IMEKO World Congress, Vol. 1, 344-350, Busan, Republic of Korea, 9-14 September 2012.
22. G. E. D'Errico, N. Murru, An algorithm for concurrent estimation of time-varying quantities, Measurement Science and Technology, Vol. 23, No. 4, 2012.
23. G. E. D'Errico, N. Murru, Fuzzy treatment of candidate outliers in measurements, Advances in Fuzzy Systems, Article ID 783843, 2012.
24. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators, Journal of Integer Sequences, Vol. 14, Article 11.8.1, 2011.
25. M. Abrate, S. Barbero, U. Cerruti, N. Murru, Accelerations of generalized Fibonacci sequences, Fibonacci Quarterly, Vol. 49, No. 3, 255-266, 2011.
26. S. Barbero, U. Cerruti, N. Murru, Transforming recurrent sequences by using the Binomial and Invert operators, Journal of Integer Sequences, Vol. 13, Article 10.7.7, 2010.
27. S. Barbero, U. Cerruti, N. Murru, Solving the Pell equation via Rédei rational functions, Fibonacci Quarterly, Vol. 48, No. 4, 348-357, 2010.
28. S. Barbero, U. Cerruti, N. Murru, Generalized Rédei rational functions and rational approximations over conics, International Journal of Pure and Applied Mathematics, Vol. 64, No. 2, 305-317, 2010.
29. S. Barbero, U. Cerruti, N. Murru, A generalization of the Binomial operator and its action on linear recurrent sequences, Journal of Integer Sequences, Vol. 13, Article 10.9.7, 2010.

Articles in press or submitted.

• N. Murru, C. Sanna, On the k-regularity of the k-adic valuation of Lucas sequences, Accepted for publication in Journal de Théorie des Nombres de Bordeaux. For integers $k \geq 2$ and $n \neq 0$, let $\nu_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $(u_n)_{n \geq 0}$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n + 1} + b u_n$, for some integers $a$ and $b$. Shu and Yao showed that for any prime number $p$ the sequence $\nu_p(u_{n + 1})_{n \geq 0}$ is $p$-regular, while Medina and Rowland found the rank of $\nu_p(F_{n + 1})_{n \geq 0}$, where $F_n$ is the $n$-th Fibonacci number. We prove that if $k$ and $b$ are relatively prime then $\nu_k(u_{n + 1})_{n \geq 0}$ is a $k$-regular sequence, and for $k$ a prime number we also determine its rank. Furthermore, as an intermediate result, we give explicit formulas for $\nu_k(u_n)$, generalizing a previous theorem of Sanna concerning $p$-adic valuations of Lucas sequences.
• T. Armano, M. Borsero, A. Capietto, N. Murru, A. Panzarea, A. Ruighi, On the accessibility of Moodle 2 by visually impaired users, with a focus on mathematical content, Accepted for publication in Universal Access in the Information Society. In this paper we study the accessibility towards visually impaired people of the Learning Management System Moodle. The study is conducted by testing four different visually impaired subjects, with different degrees of disability and performing different tasks connected to different roles in the LMS. A peculiar focus is given to the accessibility of content involving mathematics. At the end of the paper some recommendations to improve the accessibility of Moodle are given.
• U. Cerruti, N. Murru, If primes are finite, all of them divide the number one, Accepted for publication in The American Mathematical Monthly. We propose a novel proof of the infinitude of the primes based on elementary considerations about the Legendre's function.
• M. Abrate, S. Barbero, U. Cerruti, N. Murru, Groups and monoids of Pythagorean triples connected to conics, submitted to Revista de la Unión Matemática Argentina. We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 X 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.
• S. D'Antonio, F. Di Ginadomenico, C. Di Sarno, N. Murru, A Kalman filter based approach to anomaly detection in smart grids, Submitted to Transaction on Industrial Informatics. Modern societies are increasingly relying on smart grid infrastructures for vital services, thus motivating the efforts to develop advanced and sophisticated technologies capable of improving resilience of such infrastructures in case of either accidental or malicious intentional faults. In this context, accurate fault detection and diagnosis are of utmost importance since they enable a suitable treatment and mitigation of the malfunctions being detected through the implementation of appropriate and effective countermeasures. In this paper we propose an efficient Kalman filter-based approach to runtime data analysis in smart grids. The features of the developed approach, which gathers inputs on the smart grid status from a Wide Area Monitoring System, include the ability to detect anomalous measurements generated in presence of faults, to discriminate between accidental and malicious faults, and to provide accurate and reliable state estimates from measurements affected by noise. The proposed approach has been validated on an IEEE test case through the setup of a variety of simulation scenarios. The obtained experimental results are presented in this paper.
• G. Airò Farulla, N. Murru, R. Rossini, A fuzzy approach for segmentation of touching characters, Submitted to Pattern Recognition. The problem of correctly segmenting touching characters is an hard task to solve and it is of major relevance in many important fields, e.g., automatic character recognition. In the recent years, many methods and algorithms have been proposed; still, a definitive solution is far from being found. In this paper, we propose a novel method based on fuzzy logic. The proposed method combines in a novel way three features for segmenting touching characters that have been already proposed in other studies but have been exploited only singularly so far. The proposed strategy is based on a 3--input/1--output fuzzy inference system with fuzzy rules specifically optimized for segmenting touching characters in the case of latin printed and handwritten characters. The system performances are illustrated and supported by numerical examples showing an optimal capability of segmenting touching characters. Moreover, numerical results suggest that the method can be applied to many different datasets of characters by means of a convenient tuning of the fuzzy sets and rules.
• M. Abrate, S. Barbero, U. Cerruti, N. Murru, Linear divisibility sequences and Salem numbers, Publicationes Mathematicae. We study linear divisibility sequences of order 4, providing a characterization
by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1.
• M. Abrate, S. Barbero, U. Cerruti, N. Murru, Approximations of algebraic irrationalities with matrices, submitted to Hokkaido Mathematical Journal. We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that all the ratios between two entries of the matrix of the regular representation converge to specific algebraic irrationalities. As an interesting special case, we focus on cubic extensions giving a generalization of the Khovanskii matrices for approximating cubic irrationalities. We discuss the quality of such approximations considering both rate of convergence and size of denominators. Moreover, we briefly perform a numerical comparison with well--known iterative methods (such as Newton and Halley ones), showing that the approximations provided by regular representations appear more accurate for the same size of the denominator.

Borsista presso l'INRIM di Torino con una borsa di ricerca per il progetto "Software per il trattamento dati applicato alla metrologia dimensionale" da 01/06/2011 a 31/05/2012.

Assegnista presso il CNR di Pisa, ISTI (Istituto di Scienza e Tecnologie dell'Informazione), con un assegno di ricerca per il progetto "Studio di soluzioni matematiche per la caratterizzazione e ottimizzazione di riconfigurazioni di sistemi di distribuzione elettrica, a supporto della modellazione ed analisi di tali sistemi" da 01/07/2013 a 30/04/2014.

Assegnista presso Università degli Studi di Torino, Dipartimento di Matematica, con un assegno di ricerca per il progetto ""Individuazione e sviluppo di nuove tecnologie per favorire la partecipazione attiva agli studi universitari da parte di giovani con disabilita motoria e sensoriale", dal 01/05/2014 - in corso.

### Pubblicazioni

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