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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] |
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] |
Open Cover
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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Theory
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a 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, such as the boundary of a 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 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 categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrizable Space
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 (X, \mathcal) is said to be metrizable if there is a metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ... d : X \times X \to [0, \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Properties Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff space, Hausdorff paracompact spaces (and hence Normal space, normal and Tychonoff space, Tychonoff) and First-countable space, first-countable. However, some properties of the metric, such as completeness, cannot be said ... [...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] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...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] |
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] |
Infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \operatorname B^n or \oper ... [...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] |
