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

Danish Danish may refer to: * Something of, from, or related to the country of Denmark People * A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark * Culture of Denmark * Danish people or Danes, people with a Danish a ...

actuary 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, structure, space, models, and change. History On ...

who was born in Nustrup, Duchy of

Schleswig The Duchy of Schleswig ( da, Hertugdømmet Slesvig; german: Herzogtum Schleswig; nds, Hartogdom Sleswig; frr, Härtochduum Slaswik) was a duchy in Southern Jutland () covering the area between about 60 km (35 miles) north and 70 km ...

Denmark ) , song = ( en, "King Christian stood by the lofty mast") , song_type = National and royal anthem , image_map = EU-Denmark.svg , map_caption = , subdivision_type = Sovereign state , subdivision_name = Danish Realm, Kingdom of Denmark ...

and died in

Copenhagen Copenhagen ( or .; da, København ) is the capital and most populous city of Denmark, with a proper population of around 815.000 in the last quarter of 2022; and some 1.370,000 in the urban area; and the wider Copenhagen metropolitan ar ...

, Denmark. Important papers of his include ''On series expansions determined by the methods of least squares'', and ''Investigations of the number of

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

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 is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner produ ...

, 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 \operatorname(s) > ...

(the leading function in

Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...

'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 ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is t ...

). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the

zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy 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 functions, ...

s directly instead of the zeta function. In control theory, the Gramian or

Gram matrix In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...

is an important contribution named after him. The

Controllability Gramian In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymb ...

and

Observability Gramian In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is observable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol ...

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 c ...

. The Gram matrix is also important in

deep learning Deep learning (also known as deep structured learning) is part of a broader family of machine learning methods based on artificial neural networks with representation learning. Learning can be supervised, semi-supervised or unsupervised. De ...

, 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 In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', ...

, the

Gram–Charlier series The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The ...

, 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 bicycle.

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 {{denmark-scientist-stub