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This paper aimed to provide an approach to investigate the historical roots and evolution of research fields in China by extending the reference publication year spectroscopy (RPYS). RootCite, an open source software accepts raw data from both the Web of Science and the China Social Science Citation Index (CSSCI), was developed using python. We took iMetrics in China as the research case. 5,141 Chinese iMetrics related publications with 73,376 non-distinct cited references (CR) collected from the CSSCI were analyzed using RootCite. The results showed that the first CR in the field can be dated back to 1882 and written in English; but the majority (64.2%) of the CR in the field were Chinese publications. 17 peaks referring to 18 seminal works (13 in English and 5 in Chinese) were located during the period from 1900 to 2017. The field shared the same roots with that in the English world but has its own characteristics, and it was then shaped by contributions from both the English world and China. The five Chinese works have played irreplaceable and positive roles in the historical evolutionary path of the field, which should not be ignored, especially for the evolution of the field. This research demonstrated how RootCite aided the task of identifying the origin and evolution of research fields in China, which could be valuable for extending RPYS for countries with other languages., Comment: 19 pages,9 figures

Given an integer $q\ge2$ and $\theta_1,\cdots,\theta_{q-1}\in\{0,1\}$, let $(\theta_n)_{n\ge0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto0\theta_1\cdots\theta_{q-1}$$ $$1\mapsto1\overline{\theta}_1\cdots\overline{\theta}_{q-1}$$ beginning with $\theta_0:=0$, where $\overline{0}:=1$ and $\overline{1}:=0$. For rational functions $R$, we study infinite products of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{\theta_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{\theta_n}.$$ This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue-Morse sequence $(t_n)_{n\ge0}$ of the forms $$\prod_{n=1}^\infty\Big(R(n)\Big)^{(-1)^{t_n}}\quad\text{and}\quad\prod_{n=1}^\infty\Big(R(n)\Big)^{t_n}.$$

We study some divisibility properties related to the factors of the discriminant of the characteristic polynomial of generalized Fibonacci sequences $(G_n)_{n\ge0}$ defined by $G_0=0$, $G_1=1$ and $G_n=pG_{n-1}+qG_{n-2}$ for $n\ge2$, where $p,q$ are given integers. As corollaries, we give some divisibility properties on some well known sequences.

For integer $m\ge3$, we study the dynamical system $(\Lambda_m,\sigma_m)$ where $\Lambda_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain $0^m$ or $1^m\}$ and $\sigma_m$ is the shift map on $\{0,1\}^\mathbb{N}$ restricted to $\Lambda_m$, study the Bernoulli-type measures on $\Lambda_m$ and find out the unique equivalent $\sigma_m$-invariant ergodic probability measure. As an application, we obtain the Hausdorff dimension of the set of univoque sequences, the Hausdorff dimension of the set of sequences in which the lengths of consecutive $0$'s and consecutive $1$'s are bounded, and the Hausdorff dimension of their frequency subsets.

Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.

We study the dynamics of a simple random walk on subshifts defined by the beta transformation and apply it to find concrete formulae for the Hausdorff dimension of digit frequency sets for $\beta>1$ that solves $\beta^{m+1}-\beta^m-1=0$ generalising the work of Fan and Zhu. We also give examples of $\beta$ where this approach fails.

Let $\beta>1$ be a non-integer. First we show that Lebesgue almost every number has a $\beta$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $\beta$-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced $\beta$-expansions, where an infinite sequence on the finite alphabet $\{0,1,\cdots,m\}$ is called balanced if the frequency of the digit $k$ is equal to the frequency of the digit $m-k$ for all $k\in\{0,1,\cdots,m\}$. Finally we consider variable frequency and prove that for every pseudo-golden ratio $\beta\in(1,2)$, there exists a constant $c=c(\beta)>0$ such that for any $p\in[\frac{1}{2}-c,\frac{1}{2}+c]$, Lebesgue almost every $x$ has infinitely many $\beta$-expansions with frequency of zeros equal to $p$.

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non-nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of "weak records"(or "ladder points") in two kinds of one-dimensional non-nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for our cases., Comment: 17 Pages

By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $\beta>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_\beta$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$. It has been used in some former papers without proof. Then we clarify that for calculating the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit order when the $\beta$-expansion of $1$ is finite. Concretely, it suffices to optimize a function with finitely many variables under some restrictions. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $\beta$'s, which are called pseudo-golden ratios (also called multinacci numbers).

It is known that for the 2n-step symmetric simple random walk on Z, two events have the same probability if and only if their sets of paths have the same cardinality. In this article, we construct two kinds of bijections between sets of paths with the same cardinality. The construction is natural and simple. It can be easily realized through programming. More importantly, this construction opens a door to prove that two events in the 2n-step symmetric simple random walk on Z have the same probability and some further related results., Comment: 8 pages, 2 figures

In this paper, we consider the N-urn Ehrenfest model. By utilizing an auxiliary continuous-time Markov chain, we obtain the explicit formula for the Laplace transform of the hitting time from a single state to a set A of states where A satisfies some symmetric properties. After obtaining the Laplace transform, we are able to compute the high-order moments(especially, variance) for the hitting time., Comment: 18 pages

Control and manipulation of single charges and their internal degrees of freedom, such as spins, is a fundamental goal of nanoscience with promising technological applications. Recently, atomically thin semiconductors such as WSe$_2$ have emerged as a platform for valleytronics, offering rich possibilities for optical, magnetic and electrical control of the valley index. While progress has been made in controlling valley index of ensemble of charge carriers, valley control of individual charges, crucial for valleytronics, remains unexplored. Here, we provide unambiguous evidence for localized holes with net spin in optically active WSe$_2$ quantum dots (QDs) and control their spin-valley state with the helicity of the excitation laser under small magnetic field. We estimate a lower bound on the valley lifetime of a single charge in QD from recombination time to be $\sim$ nanoseconds. Remarkably, neutral QDs do not exhibit such a control, demonstrating the role of excess charge in prolonging the valley lifetime. Our work extends the field of 2D valleytronics to the level of single spin-valley, relevant for quantum information and sensing applications

Quantum entanglement is a fundamental phenomenon which, on the one hand, reveals deep connections between quantum mechanics, gravity and the space-time; on the other hand, has practical applications as a key resource in quantum information processing. While it is routinely achieved in photon-atom ensembles, entanglement involving the solid-state or macroscopic objects remains challenging albeit promising for both fundamental physics and technological applications. Here, we report entanglement between collective, chiral vibrations in two-dimensional (2D) WSe$_2$ host --- chiral phonons (CPs) --- and single-photons emitted from quantum dots (QDs) present in it. CPs which carry angular momentum were recently observed in WSe$_2$ and are a distinguishing feature of the underlying honeycomb lattice. The entanglement results from a "which-way" scattering process, involving an optical excitation in a QD and doubly-degenerate CPs, which takes place via two indistinguishable paths. Our unveiling of entanglement involving a macroscopic, collective excitation together with strong interaction between CPs and QDs in 2D materials opens up ways for phonon-driven entanglement of QDs and engineering chiral or non-reciprocal interactions at the single-photon level.

We review the results in Chen & Yao(2009,2012) which concern the contact process in a static random environment on the half space Z^d*Z^+ and make some addition to them. Furthermore, we explain why our methods cannot apply to the whole space case and compare our results with some related works., Comment: 12 pages, 4 figures