Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish

actuary An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require investment management, asset management, ...

and

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 was born in Nustrup, Duchy of

Schleswig The Duchy of Schleswig (; ; ; ; ; ) was a duchy in Southern Jutland () covering the area between about 60 km (35 miles) north and 70 km (45 mi) south of the current border between Germany and Denmark. The territory has been di ...

Denmark Denmark is a Nordic countries, Nordic country in Northern Europe. It is the metropole and most populous constituent of the Kingdom of Denmark,, . also known as the Danish Realm, a constitutionally unitary state that includes the Autonomous a ...

and died in

Copenhagen Copenhagen ( ) is the capital and most populous city of Denmark, with a population of 1.4 million in the Urban area of Copenhagen, urban area. The city is situated on the islands of Zealand and Amager, separated from Malmö, Sweden, by the ...

, Denmark. Important papers of his include ''On series expansions determined by the methods of least squares'', and ''Investigations of the number of primes less than a given number''. The mathematical method that bears his name, the

Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other. By technical definition, it is a metho ...

, was first published in the former paper, in 1883. For number theorists his main fame is the series for the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

(the leading function in Riemann's exact

prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal ...

). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the

Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function ...

s directly instead of the zeta function. In control theory, the Gramian or Gram matrix is an important contribution named after him. The Controllability Gramian and Observability Gramian are both important in the analysis of the stability of

control systems A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial co ...

. The Gram matrix is also important in

deep learning Deep learning is a subset of machine learning that focuses on utilizing multilayered neural networks to perform tasks such as classification, regression, and representation learning. The field takes inspiration from biological neuroscience a ...

, where it is used to represent the distribution of features in style transfer. Gram was the first mathematician to provide a systematic theory of the development of skew frequency curves, showing that the normal symmetric Gaussian error curve was but one special case of a more general class of frequency curves. Gram's theorem, the Gram–Charlier series, and Gram points are also named after him. He died on his way to a meeting of the Royal Danish Academy after being struck by a cyclist.

References

Notes Bibliography * 19th-century Danish mathematicians 20th-century Danish mathematicians 1850 births 1916 deaths Cycling road incident deaths Linear algebraists Pedestrian road incident deaths Road incident deaths in Denmark People from the Duchy of Schleswig {{denmark-scientist-stub