Valuation ringIn abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F such that either x or x−1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.
Weight (representation theory)In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.
Philippe Michel (economist)Philippe Michel (October 6, 1937 – July 22, 2004) was a French mathematical economist. Philippe Michel earned a PhD in Mathematics from the University of Paris VI in 1972, and became a Professor of Mathematics in 1976 at the University of Paris I. In 1993, he joined the Faculty of Economics at the University of Aix-Marseille II at GREQAM. Philippe Michel first scientific contributions were in the field of mathematics, with a focus on optimal control theory.
Philippe Michel (number theorist)Philippe Gabriel Michel (born 23 January 1969) is a French mathematician who holds the chair in analytic number theory at the École Polytechnique Fédérale de Lausanne in Switzerland. Michel was born in Lyon. He studied from 1989 to 1993 at the École normale supérieure de Cachan, and then moved to the University of Paris-Sud, where he earned a doctorate in 1995 under the supervision of Étienne Fouvry and then a habilitation in 1998. He was a professor at the University of Montpellier from 1998 to 2008, when he moved to EPFL.
Unitary groupIn mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication.
PermutationIn mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1, 2, 3}, namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).
Philippe of BelgiumPhilippe or Filip (Filip Leopold Lodewijk Maria ˈfilɪp ˈleːjoːpɔlt ˈloːdəʋɛik maːˈrijaː; Philippe Léopold Louis Marie filip leɔpɔld lwi maʁi; born 15 April 1960) is King of the Belgians. He is the eldest child of King Albert II and Queen Paola. He succeeded his father upon the latter's abdication for health reasons on 21 July 2013. He married Mathilde d'Udekem d'Acoz in 1999, with whom he has four children. Their eldest child, Princess Elisabeth, is first in the line of succession.
Louis-Michel van LooLouis-Michel van Loo (2 March 1707, Toulon – 20 March 1771, Paris) was a French painter. He studied under his father, the painter Jean-Baptiste van Loo, at Turin and Rome, and he won a prize at the Académie Royale de Peinture et de Sculpture in Paris in 1725. With his uncle, the painter Charles-André van Loo, he went to Rome in 1727–1732, and in 1736 he became court painter to Philip V of Spain at Madrid, where he was a founder-member of the Royal Academy of Fine Arts of San Fernando in 1752.
Louis Philippe ILouis Philippe I (6 October 1773 – 26 August 1850), nicknamed the Citizen King, was King of the French from 1830 to 1848, and the penultimate monarch of France. As Louis Philippe, Duke of Chartres, he distinguished himself commanding troops during the Revolutionary Wars and was promoted to lieutenant general by the age of nineteen, but he broke with the Republic over its decision to execute King Louis XVI. He fled to Switzerland in 1793 after being connected with a plot to restore France's monarchy.
Unitary matrixIn linear algebra, an invertible complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if where I is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.
ExponentiationIn mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the (power of) n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: The exponent is usually shown as a superscript to the right of the base.
P-adic numberIn number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending (possibly infinitely) to the left rather than to the right.
