326,091 research outputs found
Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals
No full textWe establish that constructive continued fraction dimension originally defined using s-gales [Nandakumar and Vishnoi, 2022] is robust, but surprisingly, that the effective continued fraction dimension and effective (base-b) Hausdorff dimension of the same real can be unequal in general.
We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. In the process, we construct an optimal lower semi-computable s-gale for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest.
We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that feasible dimension is invariant with respect to base conversion [Hitchcock and Mayordomo, 2013]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [Nandakumar and Vishnoi, 2022]. In contrast, for any 0 < ε < 0.5, we prove the existence of a real whose effective Hausdorff dimension is less than ε, but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the "non-faithfulness" of certain families of covers, investigated by Peres and Torbin [Peres and Torbin] and by Albeverio, Ivanenko, Lebid and Torbin [Albeverio et al., 2020].
We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension
Point-To-Set Principle and Constructive Dimension Faithfulness
No full textHausdorff Φ-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of Φ-dimension using the notion of constructive Φ-s-supergales. We prove a Point-to-Set Principle for Φ-dimension, through which we get Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive Φ-dimension.
A class of covering sets Φ is said to be "faithful" to Hausdorff dimension if the Φ-dimension and Hausdorff dimension coincide for every set. Similarly, Φ is said to be "faithful" to constructive dimension if the constructive Φ-dimension and constructive dimension coincide for every set. Using the Point-to-Set Principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of "faithfulness" of Cantor coverings at the Hausdorff and constructive levels are equivalent.
We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin [Albeverio et al., 2020] to derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion
Diffusive author(s), cohesive author: Analysis of S/N (1994)
No full textThis study indicates the ways in which various aspects of the author(s) are brought forth in Dumb type’s performance art, the S/N production. Previous research has suggested a non-hierarchical organization of Dumb type and the absence of a “privileged author” in Dumb type’s collaborative work, S/N. However, the results that I have investigated from member’s interviews on the creative process of S/N along with my analysis of the recorded images of S/N, indicate a different aspect of the author(s). First, S/N was created through, so to speak, the collective ideas of the members of Dumb type. Further, S/N has at least nine quotations from previous performances, installations, and printed writings, besides the work-in-progress technique. Explicating one of the “author functions” as given by Michel Foucault, each text has plural subjects of the author. However, it has been revealed from members’ interviews that Teiji Furuhashi had a decision-making role in selecting the members’ ideas within the performance. Since then, S/N has had plural subjects of creation; however, Furuhashi is one of the subjects of creation along with the “privileged author.” S/N has plural authors (diffusive authors) yet at the same time, it has a “privileged author,” Teiji Furuhashi (cohesive author)
Going Beyond Counting First Authors in Author Co-citation Analysis
No full textThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Real Numbers Equally Compressible in Every Base
No full textThis work solves an open question in finite-state compressibility posed by Lutz and Mayordomo [Lutz and Mayordomo, 2021] about compressibility of real numbers in different bases.
Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence.
Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to 1 in every base. At the other extreme, for example, every rational number has finite-state dimension equal to 0 in every base.
Generalizing this, Lutz and Mayordomo in [Lutz and Mayordomo, 2021] (see also Lutz [Lutz, 2012]) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number ξ and a compressibility ratio s ∈ (0,1) such that for every base b, the compressibility ratio of the base-b expansion of ξ is precisely s? It is conceivable that there is no such number. Indeed, some works explore "zero-one" laws for other feasible dimensions [Fortnow et al., 2011] - i.e. sequences with certain properties either have feasible dimension 0 or 1, taking no value strictly in between.
However, we answer the question of Lutz and Mayordomo affirmatively by proving a more general result. We show that given any sequence of rational numbers ⟨q_b⟩_{b=2}^∞, we can explicitly construct a single number ξ such that for any base b, the finite-state dimension/compression ratio of ξ in base-b is q_b. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between 0 and 1, as posed by Lutz and Mayordomo.
In our construction, we combine ideas from Wolfgang Schmidt’s construction of absolutely normal numbers from [Schmidt, 1961], results regarding low discrepancy sequences and several new estimates related to exponential sums
Dispelling the Myths Behind First-author Citation Counts
No full textWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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