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Ulisse Dini (14 November 1845 – 28 October 1918) was an
Italia Italy ( it, Italia ), officially the Italian Republic ( it, Repubblica Italiana, links=no ), is a country consisting of Italian Peninsula, a peninsula delimited by the Alps and List of islands of Italy, several islands surrounding it, whose ...

Italia
n
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

mathematician
and
politician A politician is a person active in party politics A political party is an organization that coordinates candidate A candidate, or nominee, is the prospective recipient of an award or honor, or a person seeking or being considered for some ...

politician
, born in
Pisa Pisa ( , or ) is a city and ''comune The (; plural: ) is a Administrative division, local administrative division of Italy, roughly equivalent to a township or municipality. Importance and function The provides essential public ser ...

Pisa
. He is known for his contribution to
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

real analysis
, partly collected in his book "''Fondamenti per la teorica delle funzioni di variabili reali''".


Life and academic career

Dini attended the
Scuola Normale Superiore The Scuola Normale Superiore (commonly known in Italy as "la Normale") is a university institution of higher education based in Pisa and Florence, Tuscany, Italy, currently attended by about 600 undergraduate and postgraduate (PhD) students. It ...
in order to become a teacher. One of his professors was
Enrico Betti Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an Italians, Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equa ...

Enrico Betti
. In 1865, a scholarship enabled him to visit
Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,175,601 residents , in an area of more than . Since the 17th century, Paris ha ...

Paris
, where he studied under
Charles Hermite Charles Hermite () FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Reso ...
as well as
Joseph Bertrand Joseph Louis François Bertrand (11 March 1822 – 5 April 1900) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...

Joseph Bertrand
, and published several papers. In 1866, he was appointed to the
University of Pisa The University of Pisa ( it, Università di Pisa, UniPi) is a public research university #REDIRECT Public university #REDIRECT Public university #REDIRECT Public university #REDIRECT Public university#REDIRECT Public university A public univer ...
, where he taught
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
and
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measu ...
. In 1871, he succeeded Betti as professor for
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
and
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
. From 1888 until 1890, Dini was ''rettore'' of the Pisa University, and of the ''
Scuola Normale Superiore The Scuola Normale Superiore (commonly known in Italy as "la Normale") is a university institution of higher education based in Pisa and Florence, Tuscany, Italy, currently attended by about 600 undergraduate and postgraduate (PhD) students. It ...
'' from 1908 until his death in 1918. He was also active as a
politician A politician is a person active in party politics A political party is an organization that coordinates candidate A candidate, or nominee, is the prospective recipient of an award or honor, or a person seeking or being considered for some ...

politician
: in 1871 he was voted into the Pisa
city council A municipal council is the legislature, legislative body of a municipality or local government area. Depending on the location and classification of the municipality it may be known as a city council, town council, town board, community council, ...
, and in 1880, he became a member of the
Italian parliament The Italian Parliament ( it, Parlamento italiano) is the national parliament of the Italian Republic Italy ( it, Italia ), officially the Italian Republic ( it, Repubblica Italiana, links=no ), is a country consisting of a peninsula delim ...
.


Honors

He has been elected honorary member of
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization An organization, or organisati ...
.


Work


Research activity

Dini worked in the field of mathematical analysis during a time when it was begun to be based on rigorous foundations. In addition to his books, he wrote about sixty papers. He proved the
Dini criterion In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by . Statement Dini's criterion states that if a periodic function ' has the property that (f(t)+f(-t))/t is Locally integrable function, l ...
for the
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
of
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
and investigated the
potential theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
and
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
of
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight Visual perce ...
s, based on work by
Eugenio Beltrami Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italy, Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the f ...
. His work on the theory of real functions was also important in the development of the concept of the measure on a set.See . The
implicit function theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is known in Italy as the Dini's theorem.


Teaching activity

One of his students was
Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, ...

Luigi Bianchi
.


Books by U. Dini


Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale
(Pisa, T. Nistri, 1880)
Lezioni di analisi infinitesimale. vol. 1
(Pisa, T. Nistri, 1907–1915)
Lezioni di analisi infinitesimale.vol. 2 part 1
(Pisa, T. Nistri, 1907–1915)
Lezioni di analisi infinitesimale.vol. 2 part 2
(Pisa, T. Nistri, 1907–1915)
Fondamenti per la teorica delle funzioni di variabili reali
(Pisa, T. Nistri, 1878)


See also

*
Dini criterion In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by . Statement Dini's criterion states that if a periodic function ' has the property that (f(t)+f(-t))/t is Locally integrable function, l ...
* Dini derivative *
Dini test In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function (mathematics), function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipsc ...
* Dini's theorem *
Dini's surface In geometry, Dini's surface is a surface (mathematics), surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: : \begin x&=a \cos ...

Dini's surface
*
Dini continuity In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemati ...
* Dini–Lipschitz criterion


Notes


References

*. "''Riemann's conditions for integrability and their influence on the birth of the concept of measure''" (English translation of title) is an article on the history of measure theory, analyzing deeply and comprehensively every early contribution to the field, starting from Riemann's work and going to the works of
Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a Germany, German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was ...
,
Gaston Darboux Jean-Gaston Darboux French Academy of Sciences, FAS Institute of France, MIF Royal Society of London, FRS Fellow of the Royal Society of Edinburgh, FRSE (14 August 1842 – 23 February 1917) was a French mathematician. Life According this b ...
,
Giulio Ascoli Giulio Ascoli (20 January 1843, Trieste Trieste ( , ; sl, Trst ; german: Triest ) is a city and seaport The Porticciolo del Cedas port in Barcola The thumb is the first digit of the hand, next to the index finger. When a person is stand ...
,
Henry John Stephen Smith Prof Henry John Stephen Smith FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * F ...

Henry John Stephen Smith
, Ulisse Dini,
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian lan ...

Vito Volterra
,
Paul David Gustav du Bois-Reymond Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a Germany, German mathematician who was born in Berlin and died in Freiburg. He was the brother of Emil du Bois-Reymond. His thesis was concerned with the mechanica ...
and
Carl Gustav Axel Harnack Carl Gustav Axel Harnack (, Dorpat (now – 3 April 1888, Dresden) was a Baltic German mathematician who contributed to potential theory. Harnack's inequality applied to harmonic functions. He also worked on the real algebraic geometry of plane ...
. *. ''Mathematics in the first half of the 20th century'' (English translation of the title) is a short survey on the development of mathematics in its various branches during the first half of the 20th century.


Further reading

*. *. *.


External links

* * {{DEFAULTSORT:Dini, Ulisse 1845 births 1918 deaths 19th-century Italian mathematicians
20th-century Italian mathematicians {{CatAutoTOC 20th-century Italian scientists, Math 20th-century mathematicians by nationality, Italian 20th-century Italian people by occupation, Mathemtaicians ...
20th-century Italian politicians Mathematical analysts People from Pisa University of Pisa faculty