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, a conformal map is a function that locally preserves angles, but not necessarily lengths.
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space is sometimes used as a synonym, especially in functional analysis.
In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.
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.
Jean, Baron Bourgain was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics.
In quantum field theory, a nonlinear σ model describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The non-linear σ-model was introduced by Gell-Mann & Lévy, who named it after a field corresponding to a spinless meson called σ in their model. This article deals primarily with the quantization of the non-linear sigma model; please refer to the base article on the sigma model for general definitions and classical (non-quantum) formulations and results.
Tian Gang is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis.
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, the Caristi fixed-point theorem generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε-variational principle of Ekeland. The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk.
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.
Mohamed Amine Khamsi is an American/Moroccan mathematician. His research interests include nonlinear functional analysis, the fixed point theory and metric spaces. In particular, he has made notable contributions to the fixed point theory of metric spaces. He graduated from the prestigious École Polytechnique in 1983 after attending the equally prestigious Lycée Louis-le-Grand in Paris, France. He completed his PhD, entitled "La propriété du point fixe dans les espaces de Banach et les espaces Metriques", at the Pierre-and-Marie-Curie University in May 1987 under the supervision of Gilles Godefroy. He then went on to visit University of Southern California and University of Rhode Island from 1987 to 1989. Since 1989 he has worked from the University of Texas at El Paso, as a full professor of mathematics since 1999.
Themistocles M. Rassias is a Greek mathematician, and a Professor at the National Technical University of Athens, Greece. He has published more than 300 papers, 10 research books and 45 edited volumes in research Mathematics as well as 4 textbooks in Mathematics for university students. His research work has received more than 18,000 citations according to Google Scholar and more than 5,500 citations according to MathSciNet. His h-index is 48. He serves as a member of the Editorial Board of several international mathematical journals.
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.
In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and Richard S. Hamilton by showing that, with respect to the Carathéodory metric on the domain, the holomorphic mapping becomes a contraction mapping to which the Banach fixed-point theorem can be applied.
In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim, and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.
Ismat Beg, FPAS, FIMA, is a Pakistani mathematician and researcher. Beg is a senior Full Professor at the Lahore School of Economics, Higher Education Commission Distinguished National Professor and an honorary full professor at the Mathematics Division of the Institute for Basic Research, Florida, US.
Edward Norman Dancer FAA is an Australian mathematician, specializing in nonlinear analysis.
Wei Biao Wu is a Chinese-born statistician. He is a professor of statistics at the University of Chicago.
Alain Goriely is a mathematician. Currently, he holds the statutory professorship (chair) of mathematical modelling at the University of Oxford, Mathematical Institute, he is director of the Oxford Centre for Industrial Mathematics (OCIAM), of the International Brain and Mechanics Lab (IBMTL) and Professorial Fellow at St Catherine's College, Oxford. At the Mathematical Institute, he was the director of external relations and public engagement, from 2013 until 2022, initiating the Oxford Mathematics series of public lectures.
