scholarly journals A Perfect Set of Reals with Finite Self-Information

2013 ◽  
Vol 78 (4) ◽  
pp. 1229-1246 ◽  
Author(s):  
Ian Herbert
Keyword(s):  
Hausdorff Dimension ◽  
Mutual Information ◽  
Kolmogorov Complexity ◽  
Packing Dimension ◽  
Specific Choice ◽  
Open Questions ◽  
Perfect Set ◽  
Definition Of ◽  
Image Position

AbstractWe examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information iswhereK(·) is the prefix-free Kolmogorov complexity. A realAis said to have finite self-information ifI (A : A)is finite. We give a construction for a perfect Π10class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals withK(σ)≤+KA(σ)+f (σ)for any given Δ20fwith a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Π10set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfyK(σ)≤+KA(σ)+f (σ)for allfin a ‘nice’ class of Δ20functions which includes all Δ20orders.

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

On Hausdorff and packing dimension of product spaces

1996 ◽  
Vol 119 (4) ◽  
pp. 715-727 ◽  
Author(s):  
J. D. Howroyd
Keyword(s):  
Hausdorff Dimension ◽  
Metric Spaces ◽  
Main Idea ◽  
Packing Dimension ◽  
Packing Measure ◽  
Product Spaces ◽  
Image Position

AbstractWe show that for arbitrary metric spaces X and Y the following dimension inequalities hold:where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.

Two definitions of fractional dimension

1982 ◽  
Vol 91 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Claude Tricot
Keyword(s):  
Hausdorff Dimension ◽  
Fractional Dimension ◽  
Definition Of ◽  
Image Position

The main properties of the Hausdorff dimension, here denoted by dim, areIn ℝp, in variance under a group Н of homeomorphisms: ∀HεH, dim О H = dim. The definition of H, introduced in (15) and (16), is recalled in § 2.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

scholarly journals Conditional Rényi Entropy and the Relationships between Rényi Capacities

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 526
Author(s):  
Gautam Aishwarya ◽  
Mokshay Madiman
Keyword(s):  
Mutual Information ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Basic Properties ◽  
Reference Measure ◽  
Definition Of ◽  
Discrete Setting

The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

Evaluation of invulnerability of scale-free networks using total information of local sub-graph

2018 ◽  
Vol 29 (08) ◽  
pp. 1850075
Author(s):  
Tingyuan Nie ◽  
Xinling Guo ◽  
Mengda Lin ◽  
Kun Zhao
Keyword(s):  
Mutual Information ◽  
Complex Network ◽  
Fundamental Problem ◽  
The Self ◽  
Practical Significance ◽  
Scale Free ◽  
Total Information ◽  
Scale Free Networks ◽  
Influential Nodes ◽  
Definition Of

The quantification for the invulnerability of complex network is a fundamental problem in which identifying influential nodes is of theoretical and practical significance. In this paper, we propose a novel definition of centrality named total information (TC) which derives from a local sub-graph being constructed by a node and its neighbors. The centrality is then defined as the sum of the self-information of the node and the mutual information of its neighbor nodes. We use the proposed centrality to identify the importance of nodes through the evaluation of the invulnerability of scale-free networks. It shows both the efficiency and the effectiveness of the proposed centrality are improved, compared with traditional centralities.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

European Working Group on Clinical Cell Analysis: Consensus Protocol for the Flow Cytometric Characterisation of Platelet Function

1998 ◽  
Vol 79 (05) ◽  
pp. 885-896 ◽  
Author(s):  
Gerd Schmitz ◽  
Gregor Rothe ◽  
Andreas Ruf ◽  
Stefan Barlage ◽  
Diethelm Tschöpe ◽  
...  
Keyword(s):  
Platelet Function ◽  
Diagnostic Testing ◽  
Consensus Protocol ◽  
Specific Flow ◽  
New Techniques ◽  
Flow Cytometric ◽  
Open Questions ◽  
European Working ◽  
Definition Of

IntroductionAn increased or disturbed activation and aggregation of platelets plays a major role in the pathophysiology of thrombosis and haemostasis and is related to cardiovascular disease processes. In addition to qualitative disturbances of platelet function, changes in thrombopoiesis or an increased elimination of platelets, (e. g., in autoimmune thrombocytopenia), are also of major clinical relevance. Flow cytometry is increasingly used for the specific characterisation of phenotypic alterations of platelets which are related to cellular activation, haemostatic function and to maturation of precursor cells. These new techniques also allow the study of the in vitro response of platelets to stimuli and the modification thereof under platelet-targeted therapy as well as the characterisation of platelet-specific antibodies. In this protocol, specific flow cytometric techniques for platelet analysis are recommended based on a description of the current state of flow cytometric methodology. These recommendations are an attempt to promote the use of these new techniques which are at present broadly evaluated for diagnostic purposes. Furthermore, the definition of the still open questions primarily related to the technical details of the method should help to promote the multi-center evaluation of procedures with the goal to finally develop standardized operation procedures as the basis of interlaboratory reproducibility when applied to diagnostic testing.

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