Hausdorff and packing dimensions of non-normal tuples of numbers: Non-linearity and divergence points

Abstract Much work has shown that differences in the timecourse of language processing are central to comparing native (L1) and non-native (L2) speakers. However, estimating the onset of experimental effects in timecourse data presents several statistical problems including multiple comparisons and autocorrelation. We compare several approaches to tackling these problems and illustrate them using an L1-L2 visual world eye-tracking dataset. We then present a bootstrapping procedure that allows not only estimation of an effect onset, but also of a temporal confidence interval around this divergence point. We describe how divergence points can be used to demonstrate timecourse differences between speaker groups or between experimental manipulations, two important issues in evaluating L2 processing accounts. We discuss possible extensions of the bootstrapping procedure, including determining divergence points for individual speakers and correlating them with individual factors like L2 exposure and proficiency. Data and an analysis tutorial are available at https://osf.io/exbmk/.

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Divergence points of self-conformal measures

Monatshefte für Mathematik ◽

10.1007/s00605-019-01283-9 ◽

2019 ◽

Vol 189 (4) ◽

pp. 735-763

Author(s):

Pei Wang ◽

Yong Ji ◽

Ercai Chen ◽

Yaqing Zhang

Keyword(s):

Divergence Points ◽

Conformal Measures

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Packing dimension and measure of homogeneous Cantor sets

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270004048x ◽

2006 ◽

Vol 74 (3) ◽

pp. 443-448 ◽

Cited By ~ 7

Author(s):

H.K. Baek

Keyword(s):

Lower Bound ◽

Explicit Formula ◽

Cantor Sets ◽

Packing Dimension ◽

Packing Measure ◽

Hausdorff Measures ◽

Packing Dimensions

For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.

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Projection theorems for box and packing dimensions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074168 ◽

1996 ◽

Vol 119 (2) ◽

pp. 287-295 ◽

Cited By ~ 22

Author(s):

K. J. Falconer ◽

J. D. Howroyd

Keyword(s):

Orthogonal Projection ◽

Packing Dimension ◽

Analytic Subset ◽

Box Counting ◽

Packing Dimensions ◽

Image Position ◽

Almost All ◽

Projection Theorems

AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

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Thermodynamic geometry of charged dilaton black holes in AdS spaces

Canadian Journal of Physics ◽

10.1139/cjp-2016-0387 ◽

2016 ◽

Vol 94 (10) ◽

pp. 1045-1053 ◽

Cited By ~ 3

Author(s):

Ahmad Sheykhi ◽

Seyed Hossein Hendi ◽

Fatemeh Naeimipour ◽

Shahram Panahiyan ◽

Behzad Eslam Panah

Keyword(s):

Black Holes ◽

Ricci Scalar ◽

Thermodynamic Quantities ◽

Geometrical Approach ◽

De Sitter ◽

Ads Black Holes ◽

Divergence Points ◽

The Stability ◽

Thermodynamical Behavior ◽

Dilaton Black Holes

It was shown that with the combination of three Liouville-type dilaton potentials, one can derive dilaton black holes in the background of anti-de-Sitter (AdS) spaces. In this paper, we further extend the study on the dilaton AdS black holes by investigating their thermodynamic instability through a geometry approach. First, we review thermodynamic quantities of the solutions and check the validity of the first law of thermodynamics. Then, we investigate phase transitions and stability of the solutions. In particular, we disclose the effects of the dilaton field on the stability of the black holes. We also employ the geometrical approach toward thermodynamical behavior of the system and find that the divergencies in the Ricci scalar coincide with roots and divergencies in the heat capacity. We find that the behavior of the Ricci scalar around divergence points depends on the type of the phase transition.

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Periodic Point at Real Axis for a Generalized 3x+1 Function

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.55-57.1670 ◽

2011 ◽

Vol 55-57 ◽

pp. 1670-1674 ◽

Cited By ~ 1

Author(s):

Shuai Liu ◽

Zheng Xuan Wang

Keyword(s):

Periodic Point ◽

Real Axis ◽

Exponential Function ◽

Periodic Points ◽

Divergence Points ◽

Fractal Character ◽

Convergence And Divergence ◽

Complex Exponential

In order to study the fractal character of representative complex exponential function just as generalized 3x+1 function T(x). In this essay, we proved that T(x) has periodic points of every period in bound (n, n+1) when n>1 in real axis. Then, we found the distribution of 2-periods points of T(x) in real axis. We put forward the bottom bound of 2-periodic point’s number and proved it. Moreover, we found the number of T(x)’s 2-periodic points in different bounds to validate our conclusion. Then, we extended the conclusion to i-periods points and find similar conclusion. Finally, we proved there exist endless convergence and divergence points of T(x) in real axis.

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Packing dimensions of sections of sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004198002977 ◽

1999 ◽

Vol 125 (1) ◽

pp. 89-104 ◽

Cited By ~ 5

Author(s):

K. J. FALCONER ◽

M. JÄRVENPÄÄ

Keyword(s):

Packing Dimension ◽

Packing Dimensions ◽

Local Projections

We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

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Strong and weak duality principles for fractal dimension in Euclidean space

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100073758 ◽

1995 ◽

Vol 118 (3) ◽

pp. 393-410 ◽

Cited By ~ 15

Author(s):

Colleen D. Cutler

Keyword(s):

Strong Duality ◽

The Other ◽

Duality Principle ◽

Weak Duality ◽

Pointwise Dimension ◽

Analytic Subset ◽

Analytic Set ◽

Borel Measures ◽

Dual Representations ◽

Packing Dimensions

AbstractTricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

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