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Generalizing a result given by Li, Miao and Wang in 2022, we study the spectrality of a class of infinite convolutions in $\mathbb{R}^d$ generated by sequences of nearly $d$-th power lattices. This allows us to easily construct spectral measures with and without compact supports in $\mathbb{R}^d$. According to a result on the relation between supports of infinite convolutions and sets of infinite sums, we systematically study the Hausdorff and packing dimensions of infinite sums of finite sets in $\mathbb{R}^d$. As an application, we give concrete formulae for the Hausdorff and packing dimensions of the supports of a class of spectral measures in $\mathbb{R}^d$ with the form of infinite convolutions generated by specific sequences of nearly $d$-th power lattices, and finally we deduce that there are spectral measures with and without compact supports of arbitrary Hausdorff and packing dimensions in $\mathbb{R}^d$.

In this paper, we systematically summarize and enhance the understanding of weak convergence and functional limits of record numbers in discrete-time random walks under Spitzer's condition, and extend these findings to $\sigma$--record numbers using similar methods. Additionally, we identify a sufficient condition for the existence of functional limits for record numbers in continuous-time random walks. Finally, we derive corresponding results for large deviations, moderate deviations, and laws of the iterated logarithm pertaining to record numbers in discrete-time random walks., Comment: 15 pages

In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is established and the regularity of the solution is characterized for the equation. In the context of the inverse problem, the uniqueness and instability are investigated on the determination of the random source. Furthermore, a reconstruction formula is provided for the phaseless Fourier modes of the diffusion coefficient in the random source, based on the variance of the boundary data. To reconstruct the time-dependent source function from its phaseless Fourier modes, the PhaseLift method, combined with a spectral cut-off regularization technique, is employed to tackle the phase retrieval problem. The effectiveness of the proposed method is demonstrated through a series of numerical experiments.

This paper presents a new method to obtain the deformation distribution on the main reflector of an antenna only by measuring the electric intensity on a spherical surface with the focal point as the center of the sphere, regardless of phase. Combining the differential geometry theory with geometric optics method, this paper has derived a deformation-intensity equation to relate the surface deformation to the intensity distribution of a spherical near-field directly. Based on the Finite difference method (FDM) and Gauss-Seidel iteration, deformation has been calculated from intensity simulated by GO and PO method, respectively, with relatively small errors, which prove the effectiveness of the equation proposed in this paper. By means of this method , it is possible to measure the deformation only by scanning the electric intensity of a single hemispherical near-field whose area is only about $1/15$ of the aperture. And the measurement only needs a plane wave at any frequency as the incident wave, which means that both the signals from the outer space satellite and the far-field artificial beacon could be used as the sources. The scanning can be realized no matter what attitude and elevation angle the antenna is in because the size and angle of the hemisphere are changeable.

Record numbers are basic statistics in random walks, whose deviation principles are not very clear so far. In this paper, the asymptotic probabilities of large and moderate deviations for numbers of weak records in right continuous or left continuous random walks are proved., Comment: A new version is posted by the coauthor in the identifier 1909.09763

Let $(t_n)_{n\ge0}$ be the well konwn $\pm1$ Thue-Morse sequence $$+1,-1,-1,+1,-1,+1,+1,-1,\cdots.$$ Since the 1982-1983 work of Coquet and Dekking, it is known that $\sum_{k

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