HAUSDORFF DIMENSION OF THE EXCEPTIONAL SETS CONCERNING THE RUN-LENGTH FUNCTION OF BETA-EXPANSION

For any real number [Formula: see text], and any [Formula: see text], let [Formula: see text] be the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. Recently, Tong, Yu and Zhao [On the length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12 (2016) 625–633] proved that for any [Formula: see text], for Lebesgue almost all [Formula: see text], [Formula: see text] In this paper, we quantify the size of the set of [Formula: see text] for which [Formula: see text] grows to infinity in a general speed. More precisely, for any increasing function [Formula: see text] with [Formula: see text] tending to [Formula: see text] and [Formula: see text], we show that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension.

EXCEPTIONAL SETS RELATED TO THE RUN-LENGTH FUNCTION OF BETA-EXPANSIONS

Let [Formula: see text] and [Formula: see text] be real numbers. The run-length function of [Formula: see text]-expansions denoted by [Formula: see text] is defined as the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. It is known that for Lebesgue almost all [Formula: see text], [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] goes to infinity. In this paper, we calculate the Hausdorff dimension of the subtle set for which [Formula: see text] grows to infinity with other speeds. More precisely, we prove that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension, where [Formula: see text] is a strictly increasing function satisfying that [Formula: see text] is non-increasing, [Formula: see text] and [Formula: see text] as [Formula: see text]. This result significantly extends the existing results in this topic, such as the results in [J.-H. Ma, S.-Y. Wen and Z.-Y. Wen, Egoroff’s theorem and maximal run length, Monatsh. Math. 151(4) (2007) 287–292; R.-B. Zou, Hausdorff dimension of the maximal run-length in dyadic expansion, Czechoslovak Math. J. 61(4) (2011) 881–888; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem, J. Math. Anal. Appl. 436(1) (2016) 355–365; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem revisited, Monatsh. Math. 182(4) (2017) 865–875; Y. Sun and J. Xu, A remark on exceptional sets in Erdős–Rényi limit theorem, Monatsh. Math. 184(2) (2017) 291–296; X. Tong, Y.-L. Yu and Y.-F. Zhao, On the maximal length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12(3) (2016) 625–633; J. Liu, and M.-Y. Lü, Hausdorff dimension of some sets arising by the run-length function of [Formula: see text]-expansions, J. Math. Anal. Appl. 455(1) (2017) 832–841; L.-X. Zheng, M. Wu and B. Li, The exceptional sets on the run-length function of [Formula: see text]-expansions, Fractals 25(6) (2017) 1750060; X. Gao, H. Hu and Z.-H. Li, A result on the maximal length of consecutive 0 digits in [Formula: see text]-expansions, Turkish J. Math. 42(2) (2018) 656–665, doi: 10.3906/mat-1704-119].

On the maximal length of consecutive zero digits of β-expansions

Let [Formula: see text] be a real number. For any [Formula: see text], let [Formula: see text] be the maximal length of consecutive zero digits between the first [Formula: see text] digits of [Formula: see text]’s [Formula: see text]-expansion. We prove that for Lebesgue almost all [Formula: see text], [Formula: see text]. Also the Hausdorff dimensions of the related exceptional sets are determined.

A New SPIHT Image Coder Based on Fast Lifting Wavelet Transform

To improve PSNR and coding efficiency, the paper proposes a method by improving algorithm according to image compression technology to pledge the real time of the image transmission and gain the high compression ratio under the image quality. The improved SPIHT image coding algorithm based on fast lifting wavelet transform presents fast lifting wavelet transform to improve transform course, because of many consecutive zero appearing in SPIHT quantification coding, adopting the simultaneous encoding of entropy and SPIHT. Entropy coding adopts run-length-changeable coding. Proved by the experiment, the method could achieve expected purpose, can apply in the image data transmission and storage of remote image surveillance systems.

Accuracy of a New Time-Resolved Step Counter in Children

Most pedometers record cumulative steps, limiting ability to assess level of physical activity or nonwear periods. The SportBrain iStep X1 has potential to overcome this limitation by recording and storing step count data in 60-s epochs. We evaluated accuracy of this instrument in children and the duration of consecutive zero step count minutes that indicated nonwear time periods. Seventeen children walked or ran on a treadmill at 2, 3, 4 and 5 miles/hour and walked around a track while wearing the SportBrain and Digiwalker SW-701 pedometer. We compared percent error in step counts for each pedometers relative to observer counts. A subsample wore a SportBrain pedometer during up to 5 days of usual activity. The SportBrain pedometer performed with acceptable accuracy at all evaluated treadmill speeds and during self-paced walking, recording steps within an average of 4% of observed step counts. It outperformed the Digiwalker, especially at slower speeds and in overweight children. During normal wear only 1% of zero count periods were more than 60 min. We conclude that the SportBrain iStep X1 pedometer provides a valid measure of step counts in short averaging times useful for assessing patterns of physical activity in population studies and periods of nonwear.

Comparison of compression ratios for ECG signals by using three time-frequency transformations

In this paper are presented compression results of ECG signal by using three time-frequency transformations: Discrete Wavelet Transform, Wavelet Packets and Modified Cosine Transform. By using transforms mentioned, samples of signals are transformed to appropriate groups of transformation coefficients. Almost all coefficients below the determined threshold are rounded to zero values and by inverse transform the similar signal to original one is created. By using run-length coder, consecutive zero value coefficients can be replaced by single value that shows how many consecutive coefficients with zero value exists. In this way small number of coefficients is stored, and compression is obtained. Depending on transform used, different number of coefficients is rounded to zero in different positions, hence the reconstructed signal is more or less similar to the original one. In general there exists measures that show how much reconstructed signal is similar to the original one, and the most used is Percentage Root mean square Difference (PRD). Comparison of compression is performed in obtaining the larger compression ratio for the smaller PRD.