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Effective Hausdorff dimension
 In Logic Colloquium ’01
, 2005
"... ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective sdimensional Hausdorff measures, similar to the effectivization ..."
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ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective sdimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTINLÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree ’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans. 1.
TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION
"... Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show tha ..."
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Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
Estimators of fractal dimension: Assessing the roughness of time series and spatial data
, 2010
"... Lies, damn lies, and dimension estimates ..."
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On Oscillationfree εrandom Sequences
, 2008
"... In this paper we discuss three notions of partial randomness or εrandomness. εrandomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [15] proved, Tadaki [22] and Calude et al. [3] proposed at least three different concepts of partial randomnes ..."
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In this paper we discuss three notions of partial randomness or εrandomness. εrandomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [15] proved, Tadaki [22] and Calude et al. [3] proposed at least three different concepts of partial randomness. We show that all of them satisfy the natural requirement that any εnonnull set contains an εrandom infinite word. This allows us to focus our investigations on the strongest one which is based on a priori complexity. We investigate this concept of partial randomness and show that it allows—similar to the random infinite words—oscillationfree (w.r.t. to a priori complexity) εrandom infinite words if only ε is a computable number. The proof uses the dilution principle. Alternatively, for certain sets of infinite words (ωlanguages) we show that their most complex infinite words are oscillationfree εrandom. Here the parameter ε is also computable and depends on the set chosen.
Theoretical Computer Science On Partial Randomness
, 2004
"... If x = x1x2 · · · xn · · · is a random sequence, then the sequence y = 0x10x2 · · · 0xn · · · is clearly not random; however, y seems to be “about half random”. Staiger [14, 15] and Tadaki [16] have studied the degree of randomness of sequences or reals by measuring their “degree of compress ..."
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If x = x1x2 · · · xn · · · is a random sequence, then the sequence y = 0x10x2 · · · 0xn · · · is clearly not random; however, y seems to be “about half random”. Staiger [14, 15] and Tadaki [16] have studied the degree of randomness of sequences or reals by measuring their “degree of compression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain
I. FINITESIZE EFFECTS ON THE CHARACTERIZATION OF FRACTAL SETS: f(α) CONSTRUCTION VIA BOX COUNTING ON A FINITE TWOSCALED CANTOR SET.
"... Uppsats för avläggande av teknisk licentiatexamen i teoretisk fysik vid Chalmers tekniska högskola. Uppsatsen presenteras vid ett seminarium fredagen den 6:e oktober 1995 kl. 13.15 i Forskarhuset Fysik, Chalmers tekniska högskola, sal F6217 Uppsatsen finns tillgänglig vid institutionen för tillämpad ..."
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Uppsats för avläggande av teknisk licentiatexamen i teoretisk fysik vid Chalmers tekniska högskola. Uppsatsen presenteras vid ett seminarium fredagen den 6:e oktober 1995 kl. 13.15 i Forskarhuset Fysik, Chalmers tekniska högskola, sal F6217 Uppsatsen finns tillgänglig vid institutionen för tillämpad fysik This thesis consists of an introduction and the following papers: