Ulisse Dini (14 November 1845 – 28 October 1918) was an

Italia Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and

politician A politician is a person who participates in Public policy, policy-making processes, usually holding an elective position in government. Politicians represent the people, make decisions, and influence the formulation of public policy. The roles ...

, born in

Pisa Pisa ( ; ) is a city and ''comune'' (municipality) in Tuscany, Central Italy, straddling the Arno just before it empties into the Ligurian Sea. It is the capital city of the Province of Pisa. Although Pisa is known worldwide for the Leaning Tow ...

. He is known for his contributions to

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

, 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 public university in Pisa and Florence, Tuscany, Italy, currently attended by about 600 undergraduate and postgraduate (PhD) students. Together with the University of Pi ...

in order to become a teacher. One of his professors was

Enrico Betti Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an 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 equations ...

. 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, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...

, where he studied under

Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite p ...

as well as

Joseph Bertrand Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician whose work emphasized number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was the son of ...

, and published several papers. In 1866, he was appointed to the

University of Pisa The University of Pisa (, UniPi) is a public university, public research university in Pisa, Italy. Founded in 1343, it is one of the oldest universities in Europe. Together with Scuola Normale Superiore di Pisa and Sant'Anna School of Advanced S ...

, where he taught

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

and

geodesy Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...

. In 1871, he succeeded Betti as professor for

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

and

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

. From 1888 until 1890, Dini was ''rettore'' of the Pisa University, and of the ''

'' from 1908 until his death in 1918. He was also active as a

: in 1871 he was voted into the Pisa

city council A municipal council is the 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, borough counc ...

and in 1880 became a member of the

Italian parliament The Italian Parliament () is the national parliament of the Italy, Italian Republic. It is the representative body of Italian citizens and is the successor to the Parliament of the Kingdom of Sardinia (1848–1861), the Parliament of the Kingd ...

Honors

He has been elected honorary member of

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...

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 f has the property that (f(t)+f(-t))/t is locally integrable In mathema ...

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 ...

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

and investigated the

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...

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. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

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 and touch, and is ...

s, based on work by

Eugenio Beltrami Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to ...

. 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 multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single functi ...

is known in Italy as Dini's theorem, not to be confused with

Dini's theorem In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. Formal statement I ...

Teaching activity

One of his students was

Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of th ...

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)

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 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 born on ...

Gaston Darboux Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician. Life According to his birth certificate, he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midn ...

Giulio Ascoli Giulio Ascoli (20 January 1843, Trieste, Austrian Empire – 12 July 1896, Milan) was a Jewish-Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868. In 1872 he became Professor of Algebra and Calcu ...

Henry John Stephen Smith Henry John Stephen Smith (2 November 1826 – 9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadratic forms, and Smith–Minkowski–Siegel mass formula in number theory. In m ...

, Ulisse Dini,

Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to Mathematical and theoretical biology, mathematical biology and Integral equation, integral equations, being one of the ...

, Paul David Gustav du Bois-Reymond 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 pla ...

. *. ''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.

External links

* * {{DEFAULTSORT:Dini, Ulisse 1845 births 1918 deaths 19th-century Italian mathematicians 20th-century Italian mathematicians 20th-century Italian politicians Italian mathematical analysts People from Pisa Academic staff of the University of Pisa