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, mathematical structure, structure, space, Mathematica ...
who worked in the field of
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 ...
. 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 Shape, 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 scale ...
,
space-filling curve
In mathematical analysis, a space-filling curve is a curve whose Range of a function, range reaches every point in a higher dimensional region, typically the unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Pea ...
s, partly covered by the larger class of
de Rham curve
In mathematics, a de Rham curve is a continuous fractal curve obtained as the image of the Cantor space, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including the Cantor ...
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 ( ; ; ; ; ) is a City status in Belgium, city and Municipalities in Belgium, municipality of Wallonia, and the capital of the Liège Province, province of Liège, Belgium. The city is situated in the valley of the Meuse, in the east o ...
, 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 () is a public university, public research university in Palermo, Italy. It was founded in 1806, and is currently organized in 12 Faculties.
History
The University of Palermo was officially founded in 1806, although it ...
, which he kept until 1891. He settled in Rome, where he stayed as a professor at the
Sapienza University 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 (mathematics), curve whose shape retains the same general pattern of Pathological (mathematics), irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fract ...
. 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'' (1894), ''Elements of infinitesimal calculus'' (1897). He proposed a possible definition of a limit of divergent sequence, known today as "Cesàro's sum," given by the
limit of the
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
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 cal ...
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)
(L. Alvano, Naples, 1905)
''Corso di analisi algebrica con introduzione al calcolo infinitesimale''(Bocca, Torino, 1894)
See also
*
Stolz–Cesàro theorem
*
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
or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum ...
*
Cesàro curve
*
Lévy C curve
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 in Italy
Academic staff of the Sapienza University of Rome
University of Liège alumni
Sapienza University of Rome alumni
Academic staff of the University of Palermo