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Wen Wu

Publications and source records attributed to Wen Wu.

159 records · Page 9Linked to original sources
arXiv2019

Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences

First, we show that the sum-free set generated by the period-doubling sequence is not $κ$-regular for any $κ\geq 2$. Next, we introduce a generalization of the period-doubling sequence, which we call the period-$k$-folding sequences. We show that the sum-free sets generated by the period-$k$-folding sequences also fail to be $κ$-regular for all $κ\geq 2$. Finally, we study the sum-free sets generated by Sturmian sequences that begin with `11', and their difference sequences.

math.CO↗
arXiv2019

Fast mmwave Beam Alignment via Correlated Bandit Learning

Beam alignment (BA) is to ensure the transmitter and receiver beams are accurately aligned to establish a reliable communication link in millimeter-wave (mmwave) systems. Existing BA methods search the entire beam space to identify the optimal transmit-receive beam pair, which incurs significant BA latency on the order of seconds in the worst case. In this paper, we develop a learning algorithm to reduce BA latency, namely Hierarchical Beam Alignment (HBA) algorithm. We first formulate the BA problem as a stochastic multi-armed bandit problem with the objective to maximize the cumulative received signal strength within a certain period. The proposed algorithm takes advantage of the correlation structure among beams such that the information from nearby beams is extracted to identify the optimal beam, instead of searching the entire beam space. Furthermore, the prior knowledge on the channel fluctuation is incorporated in the proposed algorithm to further accelerate the BA process. Theoretical analysis indicates that the proposed algorithm is asymptotically optimal. Extensive simulation results demonstrate that the proposed algorithm can identify the optimal beam with a high probability and reduce the BA latency from hundreds of milliseconds to a few milliseconds in the multipath channel, as compared to the existing BA method in IEEE 802.11ad.

eess.SP↗
arXiv2019

Spatio-temporal dynamics of turbulent separation bubbles

The spatio-temporal dynamics of separation bubbles induced to form in a fully-developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate are studied via direct numerical simulations. Two different separation bubbles are examined: one induced by a suction-blowing velocity profile on the top boundary and the other, by a suction-only velocity profile. The latter condition allows reattachment to occur without an externally imposed favourable pressure gradient and leads to a separation bubble more representative of those occurring over airfoils and in diffusers. The suction-only separation bubble exhibits a range of clearly distinguishable modes including a high-frequency mode and a low-frequency "breathing" mode that has been observed in some previous experiments. The high-frequency mode is well characterized by classical frequency scalings for a plane mixing layer and is associated with the formation and shedding of spanwise oriented vortex rollers. The topology associated with the low-frequency motion is revealed by applying dynamic mode decomposition to the data from the simulations and is shown to be dominated by highly elongated structures in the streamwise direction. The possibility of Görtler instability induced by the streamwise curvature on the upstream end of the separation bubble as the underlying mechanism for these structures and the associated low frequency is explored.

physics.flu-dyn↗
arXiv2018

On the additive complexity of a Thue-Morse like sequence

In this paper, we study the additive complexity $ρ^{+}_{\mathbf{t}}(n)$ of a Thue-Morse like sequence $\mathbf{t}=σ^{\infty}(0)$ with the morphism $σ: 0\to 01, 1\to 12, 2\to 20$. We show that $ρ^{+}_{\mathbf{t}}(n)=2\lfloor\log_2(n)\rfloor+3$ for all integers $n\geq 1$. Consequently, $(ρ_{\mathbf{t}}(n))_{n\geq 1}$ is a $2$-regular sequence.

math.CO↗
arXiv2018

Quantitative recurrence properties and homogeneous self-similar sets

Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $μ$ be the natural self-similar measure supported on $K$. For a positive function $φ$ defined on $\mathbb{N}$, we show that the $μ$-measure of the following set \begin{equation*} R(φ):=\{x\in K: |T^n x-x|<φ(n) \; \text{for infinitely many} \; n\in\mathbb{N}\} \end{equation*} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.

math.DS↗
arXiv2016

On simultaneous approximation of the values of certain Mahler functions

In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Padé approximations of the functions under consideration, then apply the functional equations to get an infinite sequence of approximations and use the numerical approximations obtained from this sequence.

math.NT↗
arXiv2016

On the abelian complexity of the Rudin-Shapiro sequence

In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function $ρ(n)$ which satisfies certain recurrence relations. As a consequence, the abelian complexity function is $2$-regular. Further, we prove that the box dimension of the graph of the asymptotic function $λ(x)$ is $3/2$ where $λ(x)=\lim_{k\to\infty}ρ(4^{k}x)/\sqrt{4^{k}x}$ and $ρ(x)=ρ(\lfloor x\rfloor)$ for any $x> 0$.

math.CO↗
arXiv2016

On linear independence measures of the values of Mahler functions

In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.

math.NT↗
arXiv2016

Accessible values of Assouad and the lower dimensions of subsets

Let $E$ be a subset of a doubling metric space $(X,d)$. We prove that for any $s\in [0, \dim_{A}E]$, where $\dim_{A}$ denotes the Assouad dimension, there exists a subset $F$ of $E$ such that $\dim_{A}F=s$. We also show that the same statement holds for the lower dimension $\dim_L$.

math.MG↗
arXiv2015

Random affine code tree fractals: Hausdorff and affinity dimensions and pressure

We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems are equal to the zero of the pressure. In particular, we do not presume the validity of the Falconer-Sloan condition or any other additional assumptions which have been essential in all the previously known results.

math.DS↗
arXiv2014

On the regular sum-free sets

Cameron introduced a bijection between the set of sum-free sets and the set of all zero-one sequences. In this paper, we study the sum-free sets of natural numbers corresponding to certain zero-one sequences which contain the Cantor-like sequences and some substitution sequences, etc. Those sum- free sets considered as integer sequences are 2-regular. We also prove that sequences corresponding to certain sum-free sets are automatic.

math.NT↗
arXiv2014

Hankel determinants of the Cantor sequence

In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the Hankel determinants, we prove that the irrationality exponent of the Cantor number, i.e. the transcendental number with Cantor sequence as its b-ary expansion, equals 2.

math.NT↗
arXiv2014

Explicit evaluations of the Hankel determinants of a Thue--Morse-like sequence

We obtain the explicit evaluations of the Hankel determinants of the formal power series $\prod_{k\geq 0}(1+Jx^{3^{k}})$ where $J={(\sqrt{-3}-1)}/2$, and prove that the sequence of Hankel determinants is an aperiodic automatic sequence taking value in $\{0, \pm 1, \pm J, \pm J^2\}$. This research is essentially inspired by the works about Hankel determinants of Thue--Morse-like sequences by Allouche, Peyrière, Wen and Wen (1998), Bacher (2006) and the first author (2013).

math.NT↗
arXiv2013

On the irrationality exponent of the regular paperfolding numbers

In this paper, improving the method of Allouche \emph{et al.} \cite{APWW98}, we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence module 2 is periodic with period 10 which answers Coon's conjecture \cite{CV12}. Then we extend Bugeaud's method \cite{Bugeaud11} to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2.

math.NT↗