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In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some nonnegative real number such that for all x and y in M,
In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
In mathematics, the Banach fixed-point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.
Élie Joseph Cartan, ForMemRS was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems, and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century.
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.
Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University.
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point, under some conditions on F that can be stated in general terms. Some authors claim that results of this kind are amongst the most generally useful in mathematics.
In mathematics, the isometry group of a metric space is the set of all bijective isometries from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space.
In mathematics, a symmetric space is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi that these two different types of definitions are equivalent.
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.
William Arthur ("Art") Kirk is an American mathematician. His research interests include nonlinear functional analysis, the geometry of Banach spaces and metric spaces. In particular, he has made notable contributions to the fixed point theory of metric spaces; for example, he is one of the two namesakes of the Caristi-Kirk fixed point theorem of 1976. He is also known for the Kirk theorem of 1964.
André Haefliger is a Swiss mathematician who works primarily on topology.
In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. More generally, the term motion is a synonym for surjective isometry in metric geometry, including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.
Luc-Normand Tellier is a Professor Emeritus in spatial economics of the University of Quebec at Montreal.
Asuman Güven Aksoy is a Turkish-American mathematician whose research concerns topics in functional analysis, metric geometry, and operator theory including Banach spaces, measures of non-compactness, fixed points, Birnbaum–Orlicz spaces, real trees, injective metric spaces, and tight spans. She works at Claremont McKenna College, where she is Crown Professor of Mathematics and George R. Roberts Fellow.
Brailey Sims is an Australian mathematician born and educated in Newcastle, New South Wales. He received his BSc from the University of New South Wales in 1969 and, under the supervision of J. R. Giles, a PhD from the same university in 1972. He was on the faculty of the University of New England (Australia) from 1972 to 1989. In 1990 he took up an appointment at the University of Newcastle (Australia). where he was Head of Mathematics from 1997 to 2000.
Olga Hadžić was a Serbian mathematician known for her work on fixed-point theorems.
References
- ↑ "Faculty profile". Archived from the original on 2011-10-02. Retrieved 2011-06-24.
