Hausdorff Besicovich dimension - ορισμός. Τι είναι το Hausdorff Besicovich dimension
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Τι (ποιος) είναι Hausdorff Besicovich dimension - ορισμός

INVARIANT
Besicovitch - Hausdorff dimension; Capacity dimension; Hausdorff Besicovitch dimension; Hausdorff-Besicovitch dimension; Hausdorff content; Hausdorff-Besikovitch dimension; Hausdorff–Besicovitch dimension
  • Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref>

Hausdorff distance         
MATHEMATICAL DISTANCE BETWEEN TWO SUBSETS OF A METRIC SPACE
Hausdorff metric; Pompeiu-Hausdorff metric; Pompeiu–Hausdorff distance; Pompeiu-Hausdorff distance; Pompeiu–Hausdorff metric; Hausdorf distance; Hausdorff semi-metric; Hausdorff convergence
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance,
Dimension (data warehouse)         
STRUCTURE THAT CATEGORIZES FACTS AND MEASURES IN A DATA WAREHOUSE
Dimension table; Dimension(data warehouse); Dimensional Role-Playing; Data dimension; Conformed dimension
A dimension is a structure that categorizes facts and measures in order to enable users to answer business questions. Commonly used dimensions are people, products, place and time.
Hausdorff–Young inequality         
BOUND ON THE NORM OF FOURIER COEFFICIENTS
Hausdorff-Young inequality; Hausdorff−Young inequality; Hausdorff-Young theorem; Hausdorff–Young theorem
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by .

Βικιπαίδεια

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 dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

More specifically, the Hausdorff dimension is a dimensional number associated with a metric space, i.e. a set where the distances between all members are defined. The dimension is drawn from the extended real numbers, R ¯ {\displaystyle {\overline {\mathbb {R} }}} , as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.