__NOTOC__
Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian
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, structure, space, models, and change.
History
On ...
who worked in the field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. He wrote a book, ''Lezioni di geometria intrinseca'' (Naples, 1890), on this topic, in which he also describes
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
,
space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
s, partly covered by the larger class of
de Rham curve
In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.
The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all spe ...
s, but are still known today in his honor as Cesàro curves. He is known also for his 'averaging' method for the 'Cesàro-summation' of
divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mus ...
, known as the
Cesàro mean.
Biography
After a rather disappointing start of his academic career and a journey through Europe - with the most important stop at
Liège
Liège ( , , ; wa, Lîdje ; nl, Luik ; german: Lüttich ) is a major city and municipality of Wallonia and the capital of the Belgian province of Liège.
The city is situated in the valley of the Meuse, in the east of Belgium, not far from b ...
, where his older brother Giuseppe Raimondo Pio Cesàro was teaching mineralogy at the local university - Ernesto Cesàro graduated from the University of Rome in 1887, while he was already part of the Royal Science Society of Belgium for the numerous works that he had already published.
The following year, he obtained a mathematics chair at the
University of Palermo
The University of Palermo ( it, Università degli Studi di Palermo) is a university located in Palermo, Italy, and founded in 1806. It is organized in 12 Faculties.
History
The University of Palermo was officially founded in 1806, although its ...
, which he kept until 1891. He settled in Rome, where he stayed as a professor at the
Sapienza University
The Sapienza University of Rome ( it, Sapienza – Università di Roma), also called simply Sapienza or the University of Rome, and formally the Università degli Studi di Roma "La Sapienza", is a public research university located in Rome, Ita ...
until his accidental death, while trying to rescue his youngest son Manlio from drowning.
Work
Cesàro's main contributions are in the field of differential geometry. ''Lessons of intrinsic geometry'', written in 1894, explains in particular the construction of a
fractal curve
A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectif ...
. After that, Cesàro also studied the "snowflake curve" of
von Koch, continuous but not differentiable in any of its points.
Among his other works are ''Introduction to the mathematical theory of infinitesimal calculus'' (1893), ''
Algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunct ...
'' (1894), ''Elements of infinitesimal calculus'' (1897). He proposed a possible definition of a divergent sequence, known today as "Cesàro's sum," given by the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
of the
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
partial terms' sum.
Books by E. Cesàro
''Lezioni di geometria intrinseca''(Naples, 1896) (trans. into German under the title ''Vorlesungen über natürliche Geometrie''; 1901, 1st edn.; 1926, 2nd edn. trans. and with an appendix by
Gerhard Kowalewski
Gerhard Kowalewski (27 March 1876 – 21 February 1950) was a German mathematician and member of the Nazi party who introduced the matrices notation.
Early life
Waldemar Hermann Gerhard Kowalewski was born March 27, 1876, in Alt Järshagen in P ...
)
(L. Alvano, Naples, 1905)
''Corso di analisi algebrica con introduzione al calcolo infinitesimale''(Bocca, Torino, 1894)
See also
*
Stolz–Cesàro theorem In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time.
The Stolz–Cesàro theo ...
*
Cesàro's theorem
*
Cesàro equation In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length () from the start of the curve to the given point. It may also be given as an equation relating the radius of curv ...
*
Cesàro mean
*
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
*
Cesàro curve
*
Lévy C curve
In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Pa ...
Notes
External links
*
{{DEFAULTSORT:Cesaro, Ernesto
1859 births
1906 deaths
Scientists from Naples
19th-century Italian mathematicians
20th-century Italian mathematicians
Differential geometers
Deaths by drowning
Sapienza University of Rome faculty