ON SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS AND HAUSDORFF DIMENSIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050140
Author(s):  
JIA LIU
Keyword(s):  
Hausdorff Dimension ◽  
Lower Limit ◽  
Infinite Series ◽  
Three Dimensions ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Upper Limit ◽  
Engel Expansions ◽  
Increasing Function

For any [Formula: see text], let the infinite series [Formula: see text] be the Engel expansion of [Formula: see text]. Suppose [Formula: see text] is a strictly increasing function with [Formula: see text] and let [Formula: see text], [Formula: see text] and [Formula: see text] be defined as the sets of numbers [Formula: see text] for which the limit, upper limit and lower limit of [Formula: see text] is equal to [Formula: see text]. In this paper, we qualify the size of the set [Formula: see text], [Formula: see text] and [Formula: see text] in the sense of Hausdorff dimension and show that these three dimensions can be different.

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

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850049 ◽  
Author(s):  
LULU FANG ◽  
KUNKUN SONG ◽  
MIN WU
Keyword(s):  
Limit Theorem ◽  
Hausdorff Dimension ◽  
Maximal Length ◽  
Length Function ◽  
Run Length ◽  
Exceptional Sets ◽  
Consecutive Zero ◽  
Dyadic Expansion ◽  
Almost All ◽  
Increasing Function

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].

scholarly journals THE EXCEPTIONAL SETS ON THE RUN-LENGTH FUNCTION OF β-EXPANSIONS

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750060 ◽  
Author(s):  
LIXUAN ZHENG ◽  
MIN WU ◽  
BING LI
Keyword(s):  
Hausdorff Dimension ◽  
Maximal Length ◽  
Length Function ◽  
Exceptional Set ◽  
Run Length ◽  
Exceptional Sets ◽  
Monotonically Increasing Function ◽  
Increasing Function

Let [Formula: see text] and the run-length function [Formula: see text] be the maximal length of consecutive zeros amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. The exceptional set [Formula: see text] is investigated, where [Formula: see text] is a monotonically increasing function with [Formula: see text]. We prove that the set [Formula: see text] is either empty or of full Hausdorff dimension and residual in [Formula: see text] according to the increasing rate of [Formula: see text].

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

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850074 ◽  
Author(s):  
MENGJIE ZHANG
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Maximal Length ◽  
Run Length ◽  
Exceptional Sets ◽  
Number Formula ◽  
Consecutive Zero ◽  
Almost All ◽  
Increasing Function

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.

SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050047 ◽  
Author(s):  
LEI SHANG ◽  
MIN WU
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Slow Growth ◽  
Power Functions ◽  
Slow Growth Rate ◽  
Exceptional Sets ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Engel Expansions ◽  
Logarithmic Functions

We are concerned with the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the digit of the Engel expansion of [Formula: see text] and [Formula: see text] is a function such that [Formula: see text] as [Formula: see text]. The Hausdorff dimension of [Formula: see text] is studied by Lü and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490–498] under the condition that [Formula: see text] grows to infinity. The aim of this paper is to determine the Hausdorff dimension of [Formula: see text] when [Formula: see text] slowly increases to infinity, such as in logarithmic functions and power functions with small exponents. We also provide a detailed analysis of the gaps between consecutive digits. This includes the central limit theorem and law of the iterated logarithm for [Formula: see text] and the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] with the convention [Formula: see text].

scholarly journals An Experimental Study of Temperature-Gradient Metamorphism

1980 ◽  
Vol 26 (94) ◽  
pp. 303-312 ◽  
Author(s):  
D. Marbouty
Keyword(s):  
Crystal Growth ◽  
Temperature Gradient ◽  
Lower Limit ◽  
Crystal Size ◽  
Simplified Model ◽  
Constant Density ◽  
Upper Limit ◽  
Cold Room ◽  
Atmosphere Temperature ◽  
Increasing Function

Abstract The present study was carried out with a view to quantifying the effects of the main parameters ot temperature-gradient metamorphism. Cold-room simulation tests showed crystal growth to be an increasing function of the temperature-gradient modulus with a lower limit of around 0.25 deg/cm. This growth also proved to be a function of temperature itself reaching a maximum at around –5°C Furthermore the shape of depth-hoar crystals was also shown to depend on temperature and to resemble approximately that obtained when crystals are formed in the atmosphere. Temperature-gradient metamorphism is observed to take place at constant density. Increase in crystal size is a decreasing function of density with a lower limit of around 150 kg/m3, below which destructive metamorphism occurs accompanied by packing (similar to ET metamorphism): the upper limit is situated at approximately 350 kg/m3 and no depth-hoar crystals occur above this value. The series of Cold-room simulations enabled a highly simplified model of crystal growth to be constructed.

On the intersections of exceptional sets in Borel’s normal number theorem and Erdös–Rényi limit theorem

2020 ◽  
pp. 1-14
Author(s):  
Mengjie Zhang
Keyword(s):  
Limit Theorem ◽  
Hausdorff Dimension ◽  
The Other ◽  
Partial Summation ◽  
Binary Expansion ◽  
Exceptional Sets ◽  
Normal Number ◽  
Almost All ◽  
Set Of Points ◽  
Increasing Function

For any [Formula: see text], let [Formula: see text] be the partial summation of the first [Formula: see text] digits in the binary expansion of [Formula: see text] and [Formula: see text] be its run-length function. The classical Borel’s normal number theorem tells us that for almost all [Formula: see text], the limit of [Formula: see text] as [Formula: see text] goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] for almost every [Formula: see text] in [Formula: see text]. In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any [Formula: see text] and [Formula: see text], we completely determine the Hausdorff dimension of the following set: [Formula: see text] where [Formula: see text] and [Formula: see text] After some minor modifications, our result still holds if we replace the denominator [Formula: see text] in [Formula: see text] with any increasing function [Formula: see text] satisfying [Formula: see text] tending to [Formula: see text] and [Formula: see text]. As a result, we also obtain that the set of points for which neither the sequence [Formula: see text] nor [Formula: see text] converges has full Hausdorff dimension.

scholarly journals An Experimental Study of Temperature-Gradient Metamorphism

1980 ◽  
Vol 26 (94) ◽  
pp. 303-312 ◽  
Author(s):  
D. Marbouty
Keyword(s):  
Crystal Growth ◽  
Temperature Gradient ◽  
Lower Limit ◽  
Crystal Size ◽  
Simplified Model ◽  
Constant Density ◽  
Upper Limit ◽  
Cold Room ◽  
Atmosphere Temperature ◽  
Increasing Function

AbstractThe present study was carried out with a view to quantifying the effects of the main parameters ot temperature-gradient metamorphism. Cold-room simulation tests showed crystal growth to be an increasing function of the temperature-gradient modulus with a lower limit of around 0.25 deg/cm. This growth also proved to be a function of temperature itself reaching a maximum at around –5°C Furthermore the shape of depth-hoar crystals was also shown to depend on temperature and to resemble approximately that obtained when crystals are formed in the atmosphere. Temperature-gradient metamorphism is observed to take place at constant density. Increase in crystal size is a decreasing function of density with a lower limit of around 150 kg/m3, below which destructive metamorphism occurs accompanied by packing (similar to ET metamorphism): the upper limit is situated at approximately 350 kg/m3 and no depth-hoar crystals occur above this value. The series of Cold-room simulations enabled a highly simplified model of crystal growth to be constructed.

PNN and BP Neural Network Fault Diagnosis Research on Electronic Accelerator Pedal Detection

2020 ◽  
Vol 14 (2) ◽  
pp. 205-220
Author(s):  
Yuxiu Jiang ◽  
Xiaohuan Zhao
Keyword(s):  
Neural Network ◽  
Fault Diagnosis ◽  
Fault Detection ◽  
Lower Limit ◽  
Bp Neural Network ◽  
Accelerator Pedal ◽  
Electronic Throttle ◽  
Measured Voltage ◽  
Upper Limit ◽  
Network Fault

Background: The working state of electronic accelerator pedal directly affects the safety of vehicles and drivers. Effective fault detection and judgment for the working state of the accelerator pedal can prevent accidents. Methods: Aiming at different working conditions of electronic accelerator pedal, this paper used PNN and BP diagnosis model to detect the state of electronic accelerator pedal according to the principle and characteristics of PNN and BP neural network. The fault diagnosis test experiment of electronic accelerator pedal was carried out to get the data acquisition. Results: After the patents for electronic accelerator pedals are queried and used, the first measured voltage, the upper limit of first voltage, the first voltage lower limit, the second measured voltage, the upper limit of second voltage and the second voltage lower limit are tested to build up the data samples. Then the PNN and BP fault diagnosis models of electronic accelerator pedal are established. Six fault samples are defined through the design of electronic accelerator pedal fault classifier and the fault diagnosis processes are executed to test. Conclusion: The fault diagnosis results were analyzed and the comparisons between the PNN and the BP research results show that BP neural network is an effective method for fault detection of electronic throttle pedal, which is obviously superior to PNN neural network based on the experiment data.

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

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