In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
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.
In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX. If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective.
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article by Tychonoff, "Über einen Funktionenraum".
In mathematics, a well-posed problem is one for which the following properties hold:
- The problem has a solution
- The solution is unique
- The solution's behavior changes continuously with the initial conditions
Pavel Sergeyevich Alexandrov, sometimes romanized Paul Alexandroff, was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topology, the Alexandroff compactification and the Alexandrov topology are named after him.
Tikhonov, sometimes spelled as Tychonoff, or Tikhonova is a Russian surname that is derived from the male given name Tikhon, the Russian form of the Greek name Τύχων, and literally means Tikhon's. It may refer to:
Sergei Petrovich Novikov is a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory. In 1970, he won the Fields Medal.
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.
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias.
The Keldysh Institute of Applied Mathematics is a research institute specializing in computational mathematics. It was established to solve computational tasks related to government programs of nuclear and fusion energy, space research and missile technology. The Institute is a part of the Department of Mathematical Sciences of the Russian Academy of Sciences. The main direction of activity of the institute is the use of computer technology to solve complex scientific and technical issues of practical importance. Since 2016, the development of mathematical and computational methods for biological research, as well as a direct solution to the problems of computational biology with the use of such methods, has also been included in the circle of scientific activities of the institute.
Tikhonov's theorem or Tychonoff's theorem can refer to any of several mathematical theorems named after the Russian mathematician Andrey Nikolayevich Tikhonov:
Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.
Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.
Alexander Andreevich Samarskii was a Soviet and Russian mathematician and academician, specializing in mathematical physics, applied mathematics, numerical analysis, mathematical modeling, finite difference methods.
Vladimir Aleksandrovich Ilyin was a Soviet and Russian mathematician, Professor at Moscow State University, Doctor of Science, Academician of the Russian Academy of Sciences who made significant contributions to the theory of differential equations, the spectral theory of differential operators, and mathematical modeling.
Charles (Chuck) William Groetsch is an American applied mathematician and numerical analyst.
Aleksei Georgievich Sveshnikov was a Russian mathematical physicist.
