HAUSDORFF DIMENSION OF THE SET CONCERNING WITH BOREL-BERNSTEIN THEORY IN L¨UROTH EXPANSIONS

Luming Shen 대한수학회 2017 대한수학회지 Vol.54 No.4

It is well known that every $x\in(0,1]$ can be expanded to an infinite L\"{u}roth series with the form of \[ x=\tfrac{1}{d_1(x)}+\cdots+\tfrac{1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots, \] where $d_n(x)\geq 2$ for all $n\geq 1$. In this paper, the set of points with some restrictions on the digits in L\"{u}roth series expansions are considered. Namely, the Hausdorff dimension of following the set $$ F_{\phi}=\{x\in (0,1]: d_n(x)\geq \phi(n), \ {\rm{i. \ o.}} \ n\} $$ is determined, where $\phi$ is an integer-valued function defined on $\mathbb{N}$, and $\phi(n)\to \infty$ as $n\to \infty$.

Molecular dynamics study of Al solute-dislocation interactions in Mg alloys

Shen, Luming Techno-Press 2013 Interaction and multiscale mechanics Vol.6 No.2

In this study, atomistic simulations are performed to study the effect of Al solute on the behaviour of edge dislocation in Mg alloys. After the dissociation of an Mg basal edge dislocation into two Shockley partials using molecular mechanics, the interaction between the dislocation and Al solute at different temperatures is studied using molecular dynamics. It appears from the simulations that the critical shear stress increases with the Al solute concentration. Comparing with the solute effect at T = 0 K, however, the critical shear stress at a finite temperature is lower since the kinetic energy of the atoms can help the dislocation conquer the energy barriers created by the Al atoms. The velocity of the edge dislocation decreases as the Al concentration increases when the external shear stress is relatively small regardless of temperature. The Al concentration effect on the dislocation velocity is not significant at very high shear stress level when the solute concentration is below 4.0 at%. Drag coefficient B increases with the Al concentration when the stress to temperature ratio is below 0.3 MPa/K, although the effect is more significant at low temperatures.

HAUSDORFF DIMENSION OF THE SET CONCERNING WITH BOREL-BERNSTEIN THEORY IN LÜROTH EXPANSIONS

Shen, Luming Korean Mathematical Society 2017 대한수학회지 Vol.54 No.4

It is well known that every $x{\in}(0,1]$ can be expanded to an infinite $L{\ddot{u}}roth$ series with the form of $$x={\frac{1}{d_1(x)}}+{\cdots}+{\frac{1}{d_1(x)(d_1(x)-1){\cdots}d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}}+{{\cdots}}$$, where $d_n(x){\geq}2$ for all $n{\geq}1$. In this paper, the set of points with some restrictions on the digits in $L{\ddot{u}}roth$ series expansions are considered. Namely, the Hausdorff dimension of following the set $$F_{\phi}=\{x{\in}(0,1]\;:\;d_n(x){\geq}{\phi}(n),\;i.o.n}$$ is determined, where ${\phi}$ is an integer-valued function defined on ${\mathbb{N}}$, and ${\phi}(n){\rightarrow}{\infty}$ as $n{\rightarrow}{\infty}$.

A NOTE ON CONTINUED FRACTIONS WITH SEQUENCES OF PARTIAL QUOTIENTS OVER THE FIELD OF FORMAL POWER SERIES

Xuehai Hu,Luming Shen 대한수학회 2012 대한수학회보 Vol.49 No.4

Let $\mathbb{F}_q$ be a finite field with q elements and $\mathbb{F}_q((X^{-1}))$ be the field of all formal Laurent series with coefficients lying in $\mathbb{F}_q$. This paper concerns with the size of the set of points $x{\in}\mathbb{F}_q((X^{-1}))$ with their partial quotients $A_n(x)$ both lying in a given subset $\mathbb{B}$ of polynomials in $\mathbb{F}_q[X]$ ($\mathbb{F}_q[X]$ denotes the ring of polynomials with coefficients in $\mathbb{F}_q$) and deg $A_n(x)$ tends to infinity at least with some given speed. Write $E_{\mathbb{B}}=\{x:A_n(x){\in}\mathbb{B},\;deg\;A_n(x){\rightarrow}{\infty}\;as\;n{\rightarrow}{\infty}\}$. It was shown in [8] that the Hausdorff dimension of $E_{\mathbb{B}}$ is inf{$s:{\sum}_{b{\in}\mathbb{B}}(q^{-2\;deg\;b})^s$ < ${\infty}$}. In this note, we will show that the above result is sharp. Moreover, we also attempt to give conditions under which the above dimensional formula still valid if we require the given speed of deg $A_n(x)$ tends to infinity.

HOW THE PARAMETER ε INFLUENCE THE GROWTH RATES OF THE PARTIAL QUOTIENTS IN GCF_ε EXPANSIONS

Zhong, Ting,Shen, Luming Korean Mathematical Society 2015 대한수학회지 Vol.52 No.3

For generalized continued fraction (GCF) with parameter ${\epsilon}(k)$, we consider the size of the set whose partial quotients increase rapidly, namely the set $$E_{\epsilon}({\alpha}):=\{x{\in}(0,1]:k_{n+1}(x){\geq}k_n(x)^{\alpha}\;for\;all\;n{\geq}1\}$$, where ${\alpha}$ > 1. We in [6] have obtained the Hausdorff dimension of $E_{\epsilon}({\alpha})$ when ${\epsilon}(k)$ is constant or ${\epsilon}(k){\sim}k^{\beta}$ for any ${\beta}{\geq}1$. As its supplement, now we show that: $$dim_H\;E_{\epsilon}({\alpha})=\{\frac{1}{\alpha},\;when\;-k^{\delta}{\leq}{\epsilon}(k){\leq}k\;with\;0{\leq}{\delta}<1;\\\;\frac{1}{{\alpha}+1},\;when\;-k-{\rho}<{\epsilon}(k){\leq}-k\;with\;0<{\rho}<1;\\\;\frac{1}{{\alpha}+2},\;when\;{\epsilon}(k)=-k-1+\frac{1}{k}$$. So the bigger the parameter function ${\epsilon}(k_n)$ is, the larger the size of $E_{\epsilon}({\alpha})$ becomes.

A NOTE ON CONTINUED FRACTIONS WITH SEQUENCES OF PARTIAL QUOTIENTS OVER THE FIELD OF FORMAL POWER SERIES

Hu, Xuehai,Shen, Luming Korean Mathematical Society 2012 대한수학회보 Vol.49 No.4

Let $\mathbb{F}_q$ be a finite field with q elements and $\mathbb{F}_q((X^{-1}))$ be the field of all formal Laurent series with coefficients lying in $\mathbb{F}_q$. This paper concerns with the size of the set of points $x{\in}\mathbb{F}_q((X^{-1}))$ with their partial quotients $A_n(x)$ both lying in a given subset $\mathbb{B}$ of polynomials in $\mathbb{F}_q[X]$ ($\mathbb{F}_q[X]$ denotes the ring of polynomials with coefficients in $\mathbb{F}_q$) and deg $A_n(x)$ tends to infinity at least with some given speed. Write $E_{\mathbb{B}}=\{x:A_n(x){\in}\mathbb{B},\;deg\;A_n(x){\rightarrow}{\infty}\;as\;n{\rightarrow}{\infty}\}$. It was shown in [8] that the Hausdorff dimension of $E_{\mathbb{B}}$ is inf{$s:{\sum}_{b{\in}\mathbb{B}}(q^{-2\;deg\;b})^s$ < ${\infty}$}. In this note, we will show that the above result is sharp. Moreover, we also attempt to give conditions under which the above dimensional formula still valid if we require the given speed of deg $A_n(x)$ tends to infinity.

HOW THE PARAMETER ε INFLUENCE THE GROWTH RATES OF THE PARTIAL QUOTIENTS IN GCFε EXPANSIONS

Ting Zhong,Luming Shen 대한수학회 2015 대한수학회지 Vol.52 No.3

For generalized continued fraction (GCF) with parameter ε(k), we consider the size of the set whose partial quotients increase rapidly, namely the set Eε(α) := {x ∈ 2 (0, 1] : kn+1(x) ≥ kn(x)α for all n ≥ 1}, where α > 1. We in [6] have obtained the Hausdorff dimension of Eε(α) when ε(k) is constant or ε(k) ~ kβ for any β ≥ 1. As its supplement, now we show that: dimH Eε(α) = {[수식] So the bigger the parameter function ε(kn) is, the larger the size of Eε(α) becomes.

Numerical study of concrete-encased CFST under preload followed by sustained service load

Gen Li,Chao Hou,Lin-Hai Han,Luming Shen 국제구조공학회 2020 Steel and Composite Structures, An International J Vol.35 No.1

Developed from conventional concrete filled steel tubular (CFST) members, concrete-encased CFST has attracted growing attention in building and bridge practices. In actual construction, the inner CFST is erected prior to the casting of the outer reinforced concrete part to support the construction preload, after which the whole composite member is under sustained service load. The complex loading sequence leads to highly nonlinear material interaction and consequently complicated structural performance. This paper studies the full-range behaviour of concrete-encased CFST columns with initial preload on inner CFST followed by sustained service load over the whole composite section. Validated against the reported data obtained from specifically designed tests, a finite element analysis model is developed to investigate the detailed structural behaviour in terms of ultimate strength, load distribution, material interaction and strain development. Parametric analysis is then carried out to evaluate the impact of significant factors on the structural behaviour of the composite columns. Finally, a simplified design method for estimating the sectional capacity of concrete-encased CFST is proposed, with the combined influences of construction preload and sustained service load being taken into account. The feasibility of the developed method is validated against both the test data and the simulation results.