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Assouad–Nagata Dimension
In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition. Definition The Assouad–Nagata dimension of a metric space is defined as the smallest integer for which there exists a constant such that for all the space has a -bounded covering with -multiplicity at most . Here ''-bounded'' means that the diameter of each set of the covering is bounded by , and ''-multiplicity'' is the infimum of integers such that each subset of with diameter at most has a non-empty intersection with at most members of the covering. This definition can be rephrased to make it more similar to that of the Lebesgue covering dimension. The Assouad–Nagata dimension of a metric space is the smallest integer for which there exists a constant such that for every , the covering of by -balls has a refinemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A Surface (mathematics), surface, such as the Boundary (mathematics), boundary of a Cylinder (geometry), cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the Euclidean plane, plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Mathematics Research Notices
The ''International Mathematics Research Notices'' is a peer-reviewed mathematics journal. Originally published by Duke University Press and Hindawi Publishing Corporation, it is now published by Oxford University Press. retrieved 2015-02-26. The Executive Editor is Zeev Rudnick (). According to the '' |
Jun-iti Nagata
was a Japanese mathematician specializing in topology. In 1956, Jun-iti Nagata earned his PhD from Osaka University under the direction of Kiiti Morita. He was the author of two standard graduate texts in topology: ''Modern Dimension Theory'' and ''Modern General Topology''. His name is attached to the Nagata–Smirnov metrization theorem The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (t ..., which was proved independently by Nagata in 1950 and by Smirnov in 1951, as well as the Assouad–Nagata dimension of a metric space, which he introduced in a 1958 article. Nagata became a professor emeritus at both Osaka Kyoiku University, where he taught for 10 years, and Osaka Electro-Communication University, where he taught for 5 years. Works * Jun-iti Nagata: ''Modern Dimension Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics..... It is published by the Mathematics Institute of the Polish Academy of Sciences. History The journal was conceived by Zygmunt Janiszewski as a means to foster mathematical research in Poland.According to and to the introduction to the 100th volume of the journal (1978, pp=1–2). These two sources cite an article written by Janiszewski himself in 1918 and titled "''On the needs of Mathematics in Poland''". Janiszewski required that, in order to achieve its goal, the journal should not force Polish mathematicians to submit articles written exclusively in Polish, and should be devoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cover (topology)
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a ''cover'' C of X is a collection of subsets \_ of X whose union is the whole space X. In this case we say that C ''covers'' X, or that the sets U_\alpha ''cover'' X. Also, if Y is a (topological) subspace of X, then a ''cover'' of Y is a collection of subsets C=\_ of X whose union contains Y, i.e., C is a cover of Y if :Y \subseteq \bigcup_U_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Covering Dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean spaces, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by open sets. In general, a topological space ''X'' can be covered by open sets, in that one can find a collection of open sets such that ''X'' lies inside of their union. The covering dimension is the smallest number ''n'' such that for every cover, there is a refinement in which every point in ''X'' lies in the intersection of no more than ''n'' + 1 covering sets. This is the gist of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Dimension
In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monograph ''Asymptotic invariants of infinite groups'' in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture. Asymptotic dimension has important applications in geometric analysis and index theory. Formal definition Let X be a metric space and n\ge 0 be an integer. We say that \operatorname(X)\le n if for every R\ge 1 there exists a uniformly bounded cover \mathcal U of X such that every closed R-ball in X intersects at most n+1 subsets from \mathcal U. Here 'uniformly bounded' means that \sup_ \operatorname(U) <\infty . We then define the ''asymptotic dimension'' |
Assouad Dimension
In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by Patrice Assouad in his 1977 PhD thesis and later published in 1979, although the same notion had been studied in 1928 by Georges Bouligand. As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embedding, embeddability problems. Definition Let (X, d) be a metric space, and let be a non-empty subset of . For , let N_(E) denote the least number of metric open balls of radius less than or equal to with which it is possible to open cover, cover the set . The Assouad dimension of is defined to be the infimum, infimal \alpha \ge 0 for which there exist positive constants and \rho so that, whenever 0 < r < R \leq \rho, the following bound holds: The int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indiana University Mathematics Journal
The ''Indiana University Mathematics Journal'' is a journal of mathematics published by Indiana University. Its first volume was published in 1952, under the name ''Journal of Rational Mechanics and Analysis'' and edited by Zachery D. Paden and Clifford Truesdell. In 1957, Eberhard Hopf became editor, the journal name changed to the ''Journal of Mathematics and Mechanics'', and Truesdell founded a separate successor journal, the ''Archive for Rational Mechanics and Analysis'', now published by Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 .... The ''Journal of Mathematics and Mechanics'' later changed its name again to the present name. The full text of all articles published under the various incarnations of this journal is available online from the journal's we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
