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Rudolf Lipschitz | |
---|---|

Rudolf Lipschitz | |

Born | |

Died | 7 October 1903 71) | (aged

Nationality | Germany |

Alma mater | University of Königsberg |

Known for | Lipschitz continuity Lipschitz integral condition Lipschitz quaternion |

Scientific career | |

Fields | Mathematics |

Institutions | University of Bonn |

Doctoral advisor | Gustav Dirichlet Martin Ohm |

Doctoral students | Felix Klein |

**Rudolf Otto Sigismund Lipschitz** (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as number theory, algebras with involution and classical mechanics.

**Germany**, officially the **Federal Republic of Germany**, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, and the Alps, Lake Constance and the High Rhine to the south. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, France to the southwest, and Luxembourg, Belgium and the Netherlands to the west.

A **mathematician** is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

**Mathematical analysis** is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

Rudolf Lipschitz was born on 14 May 1832 in Königsberg. He was the son of a landowner and was raised at his father's estate at Bönkein which was near Königsberg.^{ [1] } He entered the University of Königsberg when he was 15, but later moved to the University of Berlin where he studied with Gustav Dirichlet. Despite having his studies delayed by illness, in 1853 Lipschitz graduated with a PhD in Berlin.^{ [2] }

**Königsberg** is the name for the historic German city that is now Kaliningrad, Russia. Originally a Sambian or Old Prussian city, it then belonged to the State of the Teutonic Order, the Duchy of Prussia, the Kingdom of Prussia, the German Empire, the Weimar Republic, and Nazi Germany. After being largely destroyed in World War II by Allied bombing and the Red Army, it was annexed by the Soviet Union and its surviving inhabitants forcibly expelled. Thereafter, the city was renamed Kaliningrad. Few traces of the former Königsberg remain today.

The **University of Königsberg** was the university of Königsberg in East Prussia. It was founded in 1544 as the world's second Protestant academy by Duke Albert of Prussia, and was commonly known as the **Albertina**.

**Johann Peter Gustav Lejeune Dirichlet** was a German mathematician who made deep contributions to number theory, and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

After receiving his PhD, Lipschitz started teaching at local Gymnasiums. In 1857 he married Ida Pascha, the daughter of one of the landowners with an estate near to his father's.^{ [1] } In 1857 he earned his habilitation at the University of Bonn and remained there as a privatdozent. In 1862 Lipschitz became an extraordinary professor at the University of Breslau where he spent the following two years. In 1864 Lipschitz moved back to Bonn as a full professor, remaining there for the rest of his career. Here he examined the dissertation of Felix Klein. Lipschitz died on 7 October 1903 in Bonn.^{ [3] }

* Gymnasium*, in the German education system, is the most advanced of the three types of German secondary schools, the others being

**Habilitation** defines the qualification to conduct self-contained university teaching and is the key for access to a professorship in many European countries. Despite all changes implemented in the European higher education systems during the Bologna Process, it is the highest qualification level issued through the process of a university examination and remains a core concept of scientific careers in these countries.

The **University of Bonn** is a public research university located in Bonn, Germany. It was founded in its present form as the Rhein University on 18 October 1818 by Frederick William III, as the linear successor of the Kurkölnische Akademie Bonn which was founded in 1777. The University of Bonn offers a large number of undergraduate and graduate programs in a range of subjects and has 544 professors and 32,500 students. Its library holds more than five million volumes.

Lipschitz discovered Clifford algebras in 1880,^{ [4] }^{ [5] } two years after William K. Clifford (1845–1879) and independently of him, and he was the first to use them in the study of orthogonal transformations. Up to 1950 people mentioned "Clifford–Lipschitz numbers" when they referred to this discovery of Lipschitz. Yet Lipschitz's name suddenly disappeared from the publications involving Clifford algebras; for instance Claude Chevalley (1909–1984)^{ [6] } gave the name "Clifford group" to an object that is never mentioned in Clifford's works, but stems from Lipschitz's. Pertti Lounesto (1945–2002) contributed greatly to recalling the importance of Lipschitz's role.^{ [7] }^{ [8] }

In mathematics, a **Clifford algebra** is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As *K*-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford.

In linear algebra, an **orthogonal transformation** is a linear transformation *T* : *V* → *V* on a real inner product space *V*, that preserves the inner product. That is, for each pair *u*, *v* of elements of *V*, we have

**Claude Chevalley** was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups. He was a founding member of the Bourbaki group.

*Lehrbuch der Analysis* (two volumes, Bonn 1877, 1880); *Wissenschaft und Staat* (Bonn, 1874); *Untersuchungen über die Summen von Quadraten* (Bonn, 1886); *Bedeutung der theoretischen Mechanik* (Berlin, 1876).

- Lipschitz domain
- Lipschitz quaternion
- Lipschitz continuity
- Lipschitz distance
- Lipschitz-continuous maps and contractions
- Concave moduli and Lipschitz approximation
- Dini–Lipschitz criterion
- Dini–Lipschitz test

In mathematics, a **Lipschitz domain** is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz.

In mathematical analysis, **Lipschitz continuity**, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the *Lipschitz constant* of the function. For instance, every function that has bounded first derivatives is Lipschitz.

In mathematics, the **Dini–Lipschitz criterion** is a sufficient condition for the Fourier series of a periodic function to converge uniformly at all real numbers. It was introduced by Ulisse Dini (1872), as a strengthening of a weaker criterion introduced by Rudolf Lipschitz (1864). The criterion states that the Fourier series of a periodic function *f* converges uniformly on the real line if

**Karl Theodor Wilhelm Weierstrass** was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany and gymnastics.

**William Kingdon Clifford** was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression "mind-stuff".

**Leopold Kronecker** was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by Weber (1893) as having said, "* Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk*". Kronecker was a student and lifelong friend of Ernst Kummer.

**Gregorio Ricci-Curbastro** was an Italian mathematician born in Lugo di Romagna. He is most famous as the inventor of tensor calculus, but also published important works in other fields.

**Richard Dagobert Brauer** was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation theory.

**Heinrich Eduard Heine** was a German mathematician.

**Eduard Study**, more properly **Christian Hugo Eduard Study**, was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.

**Adolf Hurwitz** was a German mathematician who worked on algebra, analysis, geometry and number theory.

In mathematics, the **Dini** and **Dini–Lipschitz tests** are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz.

**Kurt Wilhelm Sebastian Hensel** was a German mathematician born in Königsberg.

In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.

**Albert J. Crumeyrolle** (1919–1992) was a French mathematician and professor of mathematics at the Paul Sabatier University, known for his contributions to spinor structures and Clifford algebra.

**Heinrich Eduard Schröter** was a German mathematician, who studied geometry in the tradition of Jakob Steiner.

**Karl Rudolf Fueter** was a Swiss mathematician, known for his work on number theory.

- 1 2 http://www-history.mcs.st-andrews.ac.uk/Biographies/Lipschitz.html
- ↑ McElroy, Tucker (2009).
*A to Z of Mathematicians*. Infobase Publishing. p. 176. ISBN 978-1-438-10921-3. - ↑ Chang, Sooyoung (2011).
*Academic Genealogy of Mathematicians*. World Scientific. p. 27. ISBN 978-9-814-28229-1. - ↑ R. Lipschitz (1880). "Principes d'un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions".
*C. R. Acad. Sci. Paris*.**91**: 619–621, 660–664. - ↑ R. Lipschitz (signed) (1959). "Correspondence".
*Ann. of Math*.**69**(1): 247–251. doi:10.2307/1970102. - ↑ Chevalley, Claude (1997).
*The Algebraic Theory of Spinors and Clifford Algebras*(Collected Works Vol. 2 ed.). Springer-Verlag. pp. 48, 113. ISBN 978-3-540-57063-9. - ↑ Lounesto, Pertti (1997).
*Clifford Algebras and Spinors*. Cambridge University Press. p. 220. ISBN 978-0-521-59916-0. - ↑ Jacques Helmstetter, Artibano Micali:
*Quadratic Mappings and Clifford Algebras*, Birkhäuser, 2008, ISBN 978-3-7643-8605-4 Introduction, p. ix*ff.*

- O'Connor, John J.; Robertson, Edmund F., "Rudolf Lipschitz",
*MacTutor History of Mathematics archive*, University of St Andrews . - Rudolf Lipschitz at the Mathematics Genealogy Project
- H. Kortum. "1903 Obituary".
*In Jahresbericht DMV 16*. pp. 56–59. Retrieved 16 July 2006. (digitalized document, provided without fee by Göttingen Digitalization Project, in German)

**Edmund Frederick Robertson** is a Professor emeritus of pure mathematics at the University of St Andrews.

The **MacTutor History of Mathematics archive** is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathematicians, as well as information on famous curves and various topics in the history of mathematics.

The **University of St Andrews** is a public university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland and, following Oxford and Cambridge universities, the third-oldest university in the English-speaking world. St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy.

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