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Théophile Lepage | |
---|---|

Born | Limburg | 24 March 1901

Died | 1 April 1991 90) Verviers | (aged

Nationality | Belgian |

Alma mater | Université libre de Bruxelles |

Known for | Congruence of Lepage Calculus of variations Lepagian forms |

Scientific career | |

Fields | Mathematics |

Doctoral advisor | Théophile de Donder |

**Théophile Lepage ** (24 March 1901 – 1 April 1991) was a Belgian mathematician.

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

Théophile Henri Joseph Lepage, better known as Théophile Lepage, was born in Limburg on March 24, 1901. Together with Alfred Errera he founded the seminar for mathematical analysis at the ULB. This seminar played an important role in the flourishing of the department of mathematics at this university.^{ [1] } He was professor of mathematics at the University of Liège from 1928 till 1930. He taught differential and integral calculus at the ULB from 1931 till 1956 and higher analysis from 1956 till 1971.

**Limbourg** or **Limbourg-sur-Vesdre** is a city located in the province of Liège, Wallonia, Belgium.

**Alfred Errera** was a Belgian mathematician.

The **Université libre de Bruxelles**, abbreviated **ULB**, is a French-speaking private research university in Brussels, Belgium.

For 43 years he was a member of the Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. On June 5, 1948, he was nominated a corresponding member and on June 9, 1956 an effective member of the Académie. In 1963 he became president of the Académie and director of the Klasse Wetenschappen. He was also active in the Belgisch Wiskundig Genootschap.

The **Royal Academy of Science, Letters and Fine Arts of Belgium** is the independent learned society of science and arts of the French Community of Belgium. It is also called in shorthand **Royal Academy of Belgium** (**ARB**) or *La Thérésienne* from Maria Theresa. The Dutch-speaking counterpart for the Flemish Community in Belgium is called Royal Flemish Academy of Belgium for Science and the Arts. In 2001 both academies founded a joint association for the purpose of promoting science and arts on an international level: The Royal Academies for Science and the Arts of Belgium (RASAB). All three institutions are located in the same building, the Academy Palace in Brussels.

He died in Verviers on April 1, 1991.

At the ULB, the ideas and the enthusiasm of Théophile de Donder formed the foundation of a flourishing mathematical tradition. Thanks to student Théophile Lepage, external differential calculus acquired one of the most helpful methods introduced in mathematics during the 20th century, and one for which De Donder was a pioneer, presenting new applications in the resolution of a classical problem—the partial differential equation of Monge-Ampère —and in the synthesis of the methods of Théophile de Donder, Hermann Weyl and Constantin Carathéodory into a calculus of variations of multipal integrals.

**Théophile Ernest de Donder** was a Belgian mathematician and physicist famous for his work in developing correlations between the Newtonian concept of chemical affinity and the Gibbsian concept of free energy.

In mathematics, a (real) **Monge–Ampère equation** is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function *u* of two variables *x*,*y* is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of *u* and in the second-order partial derivatives of *u*. The independent variables (*x*,*y*) vary over a given domain *D* of **R**^{2}. The term also applies to analogous equations with *n* independent variables. The most complete results so far have been obtained when the equation is elliptic.

**Hermann Klaus Hugo Weyl**, was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.

Thanks to the use of differential geometry, it is possible to avoid long and boring calculations. The results of Lepage were named in reference works. His methods are still inspiring contemporary mathematicians: Boener and Sniatycki talked about the congruence of Lepage; not so long ago, Demeter Krupka, introduced—beside the eulerian forms which correspond to the classical equations of the calculus of variations of Euler—the so-called lepagian forms^{ [2] } or equivalents of Lepage in equations of variations on fiber spaces.

**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space *E* and a product space is defined using a continuous surjective map

We also have Lepage to thank for interesting results concerning linear representations of the symplectic group, and more specifically Lepage's dissolution of an outer potency of the product of an even number of duplicates of a complex surface.

**Representation theory** is a branch of mathematics that studies abstract algebraic structures by *representing* their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

In mathematics, the name **symplectic group** can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2*n*, *F*) and Sp(*n*). The latter is called the **compact symplectic group**. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the matrices used to represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2*n*, **C**) is denoted *C _{n}*, and Sp(

**Calculus** is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

**Thomas Joannes Stieltjes** was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral.

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

**Joseph-Louis Lagrange** was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

**Mikhail Vasilyevich Ostrogradsky** was a Russian mathematician, mechanician and physicist of Ukrainian descent. Ostrogradsky was a student of Timofei Osipovsky and is considered to be a disciple of Leonhard Euler and one of the leading mathematicians of Imperial Russia.

**Jacques Salomon Hadamard** ForMemRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.

**Bhāskara** (1114–1185), was an Indian mathematician and astronomer. He was born in Bijapur in Karnataka.

**Hilbert's nineteenth problem** is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a "*regular variational problem*" identifies precisely a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients. Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution function inherits the relatively simple and well understood structure from the solved equation.

The character **∂** is a stylized cursive *d* mainly used as a mathematical symbol to denote a partial derivative such as .

**Pierre de Fermat** was a French lawyer at the *Parlement* of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' *Arithmetica*.

**Henri Claudius Rosarius Dulac** was a French mathematician.

**Operational calculus**, also known as **operational analysis**, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.

A timeline of **calculus** and **mathematical analysis**.

**Alexis Fontaine**, known as **Alexis Fontaine des Bertins** was a French mathematician. He was a patron and teacher of Jean-Jacques de Marguerie.

In mathematical physics, the **De Donder–Weyl theory** is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theory in the way that a field is represented as a system that varies both in space and in time. This generalization is different from the canonical Hamiltonian formalism in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.

**Giuseppe Mingione** is an Italian mathematician who is active in the fields of partial differential equations and calculus of variations.

**Gianni Dal Maso** is an Italian mathematician who is active in the fields of partial differential equations, calculus of variations and applied mathematics.

**Paolo Marcellini** is an Italian mathematician who deals with mathematical analysis. He is a full professor at the University of Florence. He is the Director of the Italian National Group GNAMPA of the Istituto Nazionale di Alta Matematica Francesco Severi (INdAM).

- ↑ Mawhin, Jean, in: Robert Halleux, Geert Vanpaemel, Jan Vandersmissen en Andrée Despy-Meyer (eds.), (2001),
*Geschiedenis van de wetenschappen in België 1815-2000*,**1**, Brussel: Dexia/La Renaissance du livre, p. 71 and p. 75 - ↑ D. Krupka (1977). "A map associated to the Lepagian forms on the calculus of variations in fibred manifolds".
*Czechoslovak Mathematical Journal*.**27**(1): 114–117,118.

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