Ingrid CARBONE
Ricercatori Universitari
Analisi matematica (MATH-03/A)
Contacts
- Dipartimento di Matematica e Informatica
- Dipartimento di Matematica e Informatica
- ingrid.carbone@unical.it
- i.carbone@unical.it
- 0984/496425
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- On July 22, 1992: Graduation (summa cum laude) in Mathematics at the University of Calabria in three years and a half (instead of four).
- From November 20, 1997 to November 30, 2000: Assistant professor in mathematical analysis - Faculty of Science - University of Bari
December 1, 2000 Assistant professor in Mathematical Analisys - Faculty of Economics - University of Calabria.
- From February, 2004 to October, 2007: President of the Scientific Library of the University of Calabria.
- Currently in the board of the Ph. D. in Mathematics and Computer Science and of the Ph. D. School in Systems Engineering, Mathematics, Computer Science and Operation Research of University of Calabria.
Teaching
- From 1994 to 1997: Mathematical Analysis, Numerical Analysis, Medical Statistics - University of Calabria
- From 1997 10 2000: Advanced Analysis, Analysis 1 and Analysis 2 - University of Bari.
- From 2001: Mathematics, Calculus 1, Calculus 2, Calculus 3, Calculus 4, Analysis 1, Analysis 2, Geometry and Linear Algebra (and also Mathematics in the Faculty of Pharmacy).
Memberships
- Italian Mathematical Society since 1992.
- National Group of Functional Analysis and Applications (CNR) until 1999.
- National Group of Mathematical Analysis, Probability and Their Applications (Istituto Nazionale di Alta Matematica).
Research fellowship for young researchers from the University of Calabria (2001-2002).
Talks and conferences
- Invited lectures: Fields Institute for Research in Mathematical Sciences, Toronto, University of Warsaw; University of Linz; Graz University of Technology;
- Talks during International Meetings (Maratea, Ancona, Cluj-Napoca, Blaubeuren, Bari, Strobl, Bologna)
Current research
Uniform distribution and discrepancy, generalization of Kakutani's splitting procedure in one dimension and and higher, LS-sequences of partitions and points their discrepancy, low discrepancy sequences.
Supervisor of master and PhD thesis.
Current position: Assistant professor
Current address: University of Calabria – Department of Mathematics – Ponte P. Bucci, Cubo 30 B – 87036 Arcavacata di Rende – Cosenza - ITALY
Telephone: +39 0984 496425
E-mail: i.carbone@unical.it
Web page:
http://www.mat.unical.it/pers/docenti/carbonei/index.html
I graduated (summa cum laude) in Mathematics at the University of Calabria on July 22, 1992, in three years and a half (instead of four).
I become assistant in mathematical analysis it the Faculty of Science of the Univeristy of Bari on November 20, 1997. On December 1, 2000, I moved to the Faculty of Economics of the University of Calabria.
Since February 4, 2001, I am assistant professor in mathematical analysis in the Faculty of Economics of the University of Calabria.
PREVIOUS POSITIONS
Academic year 1996/97: Lecturer at the University of Calabria
Faculty of Pharmacy: Teaching Mathematics and Medical Statistics.
ACADEMICAL DUTIES
February 2004 - October 2007: President of the Scientific Library of the University of Calabria.
From 2008 member of the Ph. D. board in Mathematics and Computer Science and of the Ph. D. School in Systems Engineering, Mathematics, Computer Science and Operation Research at the University of Calabria.
TEACHING DUTIES
Academic year 1997/1998-1998/1999-1999/2000 at the University of Bari:
Faculty of Science: Courses on Advanced Analysis and Analysis 1 and 2.
From 2001 at the University of Calabria:
Faculty of Economics: Courses on Calculus, Analysis 1, Analysis 2, Geometry and Linear Algebra.
Faculty of Pharmacy: Course on Mathematics.
From 2013: Mathematics for Technology in Biological Sciences
PARTECIPATION TO LOCAL RESEARCH PROJECTS (EX 60%)
University of Bari:
1998 ``Topological structures, Functional analysis methods in operator theory"
1999 and 2000 ``Topological structures, Functional analysis methods in operator theory and problems on differential equations"
University of Calabria:
2001-2002-2003: ``Fixed points, operators and problems in differential equations and approximation"
2004: ``Problems in analysis"
2005-2006-2007: ``Functional analysis and applications"
2008-2009-2010-2011-2012: ``Functional analysis, probability and applications"
MEMBERSHIPS
I am member of Italian Mathematical Society since 1992.
I was member of the National Group of Functional Analysis and Applications (CNR) until 1999.
I am member of the National Group of Mathematical Analysis, Probability and Their Applications (Istituto Nazionale di Alta Matematica).
I am in the board of the Ph. D. in Mathematics and Computer Science and of the Ph. D. School in Systems Engineering, Mathematics, Computer Science and Operation Research.
GRANTS AND FELLOWSHIPS
- Research fellowship for young post graduate students from Istituto Nazionale di Alta Matematica ”F. Severi” in Rome (academic year 1992/93).
- Research fellowship for young post graduate students from Istituto Nazionale di Alta Matematica ”F. Severi” in Rome (academic year 1993/94).
- Grant from CNR (Consiglio Nazionale delle Ricerche) for research in the Department of Mathematics of the University of Bari - Supervisor: Francesco ALtomare (March 1, 1995 - February 15, 1996).
- Research fellowship for young researchers from the University of Calabria (June 18, 2001-June 17, 2002).
COLLOQUIA
- Graz University of Technology, June 15, 2012
Title: “LS-sequences of points in the unit interval” (visible on the web site of the Doctoral program in Discrete Mathematics).
- University of Linz, June 11, 2012
Title: “LS-sequences of partitions and points in dimension one and two”.
- Fields Institute for Research in Mathematical Sciences, Toronto, May 2, 2012
Title: “A new class of low discrepancy sequences of partitions and points” (visible on the web site of Fields Institute).
- University of Warsaw, Department of Mathematics, October 12, 2009 Title: “A generalization of Kakutani’s splitting procedure I, II”.
COMMUNICATIONS
-International Conference on Monte Carlo and Quasi Monte Carlo Methods (Lueven, Belgium, April 2014)
Title: “LS-sequences of points”
- International Conference and Summer School on Numerical Computation: Theory and Applications (Falerna, CS, June 2013)
Title: “Dynamical systems and low discrepancy sequenze”
- XIX Congress of the Italian Mathematical Society (Bologna, Italy, September 2011)
Title: “LS-successioni di punti a bassa discrepanza”.
- Second International Conference on Uniform Distribution (Strobl, Austria - July 2010)
Title: “Discrepancy of LS-sequences of partitions”.
- XVIII Congress of the Italian Mathematical Society (Bari, Italy, September 2007)
Title: “Successioni di partizioni e un teorema alla von Neumann”.
- Tulka Internet Seminar 2000 (Blaubeuren, Germany 2000)
Title: “Differential operators, semigroups and positive linear operators on unbounded intervals”.
- Sèminaire Itinerant d’Equations Fonctionnelles, Approximation et Convexité (Cluj-Napoca, Romania - 1996)
Title: “Shape preserving properties of some positive linear operators on unbounded intervals”.
- 3rd International Conference on Functional Analysis and Approximation Theory (Maratea, Potenza, Italy 1996)
Title: “On a new sequence of positive linear operators on unbounded intervals”
- Analisi Funzionale e Applicazioni (Ancona, Italy - 1996)
Title: “On some degenerate differential operators on weighted function spaces”
- Functional Analysis: Methods and Applications (Maratea, Potenza, Italy - 1995)
Title: “Approximation processes and diffusion equations on unbounded
intervals”
Moreover, I have taken part to several international and national meetings (apart from the ones mentioned before) in Firenze, Paris, Salzburg, Erlangen, Budapest, Stockholm, Tel Aviv, Wien, to Schools of Measure Theory in Grado and Mondello, to serveral meetings on Funcqional Analysis and Measure Theory in Ischia, Ancona, Maratea, Cefalù, Gargnano, Varenna, to UMI and UMI-CIIM meetings, to the joint meeting UMI-DMV (Perugia, 2007).
I also gave several talks at the Department of Mathematics of the University of Bari and of Calabria.
EDITORIAL BOARDS
From August 2012 - Reviewer for Mathematical Reviews/MathSciNet - Reviewer number: 080868
PREVIOUS RESEARCHES
At the beginning I was interested in complex analysis and, in particular, in Bergman spaces, and I wrote my first paper [13] on this subject.
After this period, my interests have been devoted to positive operators, approximation theory and diffusion equations. The technique and the theory used in that researches belong to Functional Analysis, Approximation Theory, Semigroup Theory of Operators and Probability Theory.
More precisely, the aim of these researches was to study some qualitative properties of the solutions of certain abstract Cauchy problems associated to degenerate differential operators of elliptic type on a class of function spaces of continuous functions defined on unbounded intervals. This analysis has been done by using strongly continuous semigroups of operators, their representation and their approximation by iterates of positive linear operators defined ad hoc. In this way it was possible to generalize previous results obtained in the context of differential operators defined on bounded intervals.
In this fields I wrote the six papers [12], [11], [10], [9], [8], [7].
These researches have been used and generalized by a group of mathematicians of the Department of Mathematics of the University of Bari and of the University of Potenza.
TEMPORARY INTERRUPTION OF RESEARCH
When, at the end of the year 2000, I moved from the University of Bari to the University of Calabria, the scientific collaboration with the local group in Bari has been interrupted. The article [7], published in 2002, can be considered the natural conclusion of that field of research.
Soon after my arrival at the University of Calabria, I was asked by the dean of the Department of Mathematics to become a member of the scientific board of the Scientific Library.
I accepted, but this duty revealed itself a full-time occupation.
In fact, after I have discovered many financial irregularities, the president had to resign and I became the President from February, 2004 till September, 2007.
During those years, I devoted all my efforts to the library, trying to solve very difficult questions, among which I think it is necessary to mention the most important one: a local provider received about 600.000 euro, but the library received no journals. I tried to force a legal action, but I received hostilities from the administration. At last, when I realized that all my efforts were useless, I decided
to resign and restart my research on a new field: uniform distribution and number theory.
During that period, I had some teaching duties, which I concluded with difficulty, and I had no time for anything else: this is the explanation for the absence of publications during the years 2003-2004-2005 and 2006.
CURRENT RESEARCHES
After the forced interruption described above, I began to study uniformly distributed sequences of partitions together with Aljoˇsa Volˇciˇc. The aim was to revive the interest on an old idea of Kakutani, which has not been fully exploited.
Also, we wanted to find a connection to the well known classical results on uniformly distributed sequences of points. After Volˇciˇc's paper “A generalization of Kakutanis splitting procedure", in which he introduced a new class of uniformly distributed sequences of partitions, called ρ-refinements, we extended Kalutani’s
procedure to higher dimension [6].
In the paper [5], written in collaboration with A. Volˇciˇc, we proved a von Neumann theorem for uniformly distributed sequences of partitions.
The paper [4] has been submitted for publication in 2010 in a different form: the title, in fact, was “Discrepancy of LS-sequences of partitions” (this version is still visible on arXiv.org). The referee asked me to add to it some result on sequences of points. In this paper I studied a family of uniformly distributed
sequences of partitions in the interval [0, 1[, that is obtained by successive refinements of [0, 1[. The paper provided precise estimates of the discrepancy of these sequences, among which there is a family having low discrepancy (which means of order 1/n). In the same paper I presented an algorithm which associates to each sequence of partitions a uniformly distributed sequence of points and I gave
estimates of their discrepancy, proving that to each sequences of partitions having low discrepancy the algorithm associates a sequences of points having low discrepancy, too (i.e., of order log n/n). In the paper it is presented, for the first time in the literature, a precise estimate for the discrepancy of a Kakutani sequence.
This paper originated several investigation in number theory. The on-line version has been already cited in different papers, while the printed version has the MathSciNet reference MR2993975.
Due to the interest the LS-sequences of points with low discrepancy have in Quasi Monte-Carlo Methods, I have studied a different form (simpler for a computer) for the algorithm which associates to each LS-sequence of partition an LS-sequence of points in terms of the representation of a infinite subset of natural numbers in base L + S. The paper [3] has been submitted for publication and is visible on arXiv.org.
In the paper [2], written in collaboration with M. R. Iac`o and A. Volˇciˇc, has been recently accepted by the journal on Ergodic Theory and Dynamical Systems. Here we focused our attention to the ergodic properties of the Kakutani sequence associated to the inverse of the
golden ratio. This sequence of partition is also the LS-sequence corresponding to L = S = 1 and has been called the Kakutani-Fibonacci sequence. By means of the well-known cutting-stacking technique, we associated to this sequence an ergodic transformation, which gives as the orbit of the origin a sequence of points which is not only uniformly distributed, but also a low discrepancy one. This paper is also visible on arXiv.org.
The preprint [1], written in collaboration with M. R. Iac`o and A. Volˇciˇc, submitted to Uniform Distribution Theory and visible on arXiv.org, is the first step towards the generalization of the LS-sequences to higher dimensions. We introduced a new definition of sequences in the unit square, which has received the attention of several mathematicians working abroad. Two of them, actually, very recently have submitted for publication
a paper dealing with the uniform distribution of these sequences (see Aistleitner-Hofer on arXiv.org).
FUTURE RESEARCHES
The introduction of LS-sequences of partitions and points revealed its interest in several fields. I mention here only some of them.
Further investigation are under development, also with M. R. Iac`o, Maria Infusino (who is currently in Reading - UK - with a grant from Marie Curie) and A.
Volˇciˇc.
The next step in ergodic theory is to associate to any LS-sequence an ergodic transformation and give its analytic expression. It will be a forthcoming paper.
Another promising research on which I am already working is the two dimensional extension of the splitting procedure using the LS-sequences of partitions. The uniform distribution has been already proved, and I am trying to compute their discrepancy.
In dimension two or more, I will also study the behavior of the LS-sequences of points in the square and, more generally, in the hypercube, trying to find a similar result of that obtained by Halton in the hypercube for the sequences of points of the van der Corput type.
SUPERVISION OF MASTER AND PH.D. THESIS
Connected with the last researches, in 2010 I was one of the supervisors (with A. Vol\v{c}i\v{c}) of Dr Aldo Rota who wrote his master thesis in Mathematics on ``Partizioni uniformemente distribuite sull'ipercubo". Graduated summa cum laude. Afterwards Dr Rota got his Ph.D. at the University of Reading (UK).
In 2011 I was one of the supervisors (with A. Vol\v{c}i\v{c}) of Dr Maria Rita Iac\`o, who wrote her master thesis in Mathematics on ``LS-successioni di punti nel quadrato". Graduated summa cum laude.
Between 2011 and 2014 I was the supervisor of Dr. M. R. Iac\`o, who is a Ph.D. student of the University of Calabria and the University of Graz due to an International Joint Doctoral Supervision (cotutelle) between the two universities. The supervisors were also A. Vol\v{c}i\v{c}) at the University of Calabria and Prof. Robert Tichy, who is one of the most famous expert in the field of research we are involved with, at the University of Graz.
In 2015 I was the one of the supervisors (with A. Vol\v{c}i\v{c}) of Dr Rossella Marazita, who wrote her master thesis in Mathematics on "Sistemici dinamici per le LS-successioni di punti". Graduated summa cum laude.
MUSICAL CAREER
1989: Summer School at the Accademia Musicale Pescarese with Sergio Cafaro
1990: Degree in Piano at the Conservatory of Cosenza with Francesco Monopoli
1992-1994: Biennal International Master Class at the Accademica Curci with Hector Pell
2001: Master Class at the Accademia ``Il trillo" with Lazar Berman
2002: Summer School at the Mozarteum of Salzburg with Aquiles delle Vigne
2002-2004: Composition and piano with Eduardo Ogando in Rome
2015: First Prize – International Music Competition “Erik Satie” Lecce (Italy)
2015: Second Prize – International Piano Competition “Città di Rocchetta” (Italy)
2015: Prize “Most distinguished musicians” from IBLA New York Foundation in 2015 IBLA Grand Prize
From 2007: Duo violin-piano with Eugenio Prete
Repertoire as soloist and chamber music.
Performances in Italy, Germany and Austria
PUBLICATIONS
[13] Alcune osservazioni sulla norma essenziale dell'operatore di composizione interna nello spazio di Bergman, Istituto Lombardo (Rend. Sci.) A 128 (1994), pp. 19-29.
[12] On a new sequence of positive linear operators on unbounded intervals (con F. Altomare), Rend. Circ. Mat. Palermo Serie II, Suppl. 40 (1996), 23-36.
[11] On some degenerate differential operators on weighted function spaces (con F. Altomare), J. Math. Analysis Appl. 213 (1997), 308-333.
[10] Shape preserving properties of some positive linear operators on unbounded intervals, J. Approx. Theory 93 No.1 (1998), 140-156.
[9] Approximation properties of some sequences of positive linear operators on weighted function spaces, Rend. Circ. Mat. Palermo, Serie II, Suppl. 52 (1998), 305-336.
[8] Markov processes and diffusion equations on unbounded intervals (con F. Altomare), Taiwanese J. of Math 5 No.1 (2001), 141-167.
[7] Trotter's type theorems for generalized Szàsz-Mirakjan operators in polynomial weighted function spaces, Indian J. of Math. 44 No.1 (2002), 1-20.
[6] Kakutani’s splitting procedure in higher dimension (con A. Volcic), Rend. Ist. Mat. Trieste 39 (2007), 119-126.
[5] A von Neumann theorem for uniformly distributed sequences of partitions (with A. Volcic). Rend. Cir. Mat. Palermo 60 n.1-2 (2011), 83-88.
[4] Discrepancy of LS-sequences of partitions and points. Ann. Mat. Pura Appl. Vol. 191, p. 819-844, ISSN: 0373-3114, doi: 10.1007/s10231-011-0208-z, MR2993975.
[3] A van der Corput-type algorithm for LS-sequences of points, submitted, arXiv:1209.3611.
[2] A dynamical system approach to the Kakutani-Fibonacci sequences (con M. R. Iacò e A. Volcic), to appear in Ergodic theory and Dynamical Systems, arXiv:1211.0708.
[1] (con M. R. Iacò e A. Volcic) LS-sequences of points in the unit square, preprint, arXiv:1211.2941.
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Contacts
- Dipartimento di Matematica e Informatica
- Dipartimento di Matematica e Informatica
- ingrid.carbone@unical.it
- i.carbone@unical.it
- 0984/496425