Multiple expansions of real numbers with digits set {0,1,q}\{0,1,q\}

For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let $\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in \mathcal{B}_k$ let $\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that $$\mathcal{B}_{\aleph_0}=[2,\infty),\quad \mathcal{B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2,$$ where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in\mathcal{B}_{k}$ the Hausdorff dimensions of $\mathcal{U}_q^{(k)}$ and $\mathcal{U}_q$ are the same, i.e., $$\dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2.$$ Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.Comment: 15 page, to appear in Mathematische Zeitschrif

Biosensors in Fermentation Applications

Biosensing technology offers new analytic routes to the use and study of fermentations, taking advantage of the high selectivity and sensitivity of the bioactive elements it exploits. Various biosensors had been commercially available today; they provide fermentation processes with convenient, accurate, and cost-effective ways of monitoring for key biochemical parameters. In this chapter, the basic ideas and principles of biosensors, especially applications of the most popular biosensors related to fermentations were highlighted

Intersections of homogeneous Cantor sets and beta-expansions

Let $\Gamma_{\beta,N}$ be the $N$-part homogeneous Cantor set with $\beta\in(1/(2N-1),1/N)$. Any string $(j_\ell)_{\ell=1}^\N$ with $j_\ell\in\{0,\pm 1,...,\pm(N-1)\}$ such that $t=\sum_{\ell=1}^\N j_\ell\beta^{\ell-1}(1-\beta)/(N-1)$ is called a code of $t$. Let $\mathcal{U}_{\beta,\pm N}$ be the set of $t\in[-1,1]$ having a unique code, and let $\mathcal{S}_{\beta,\pm N}$ be the set of $t\in\mathcal{U}_{\beta,\pm N}$ which make the intersection $\Gamma_{\beta,N}\cap(\Gamma_{\beta,N}+t)$ a self-similar set. We characterize the set $\mathcal{U}_{\beta,\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $t\in\mathcal{S}_{\beta,\pm N}$. Using techniques from beta-expansions, we show that there is a critical point $\beta_c\in(1/(2N-1),1/N)$, which is a transcendental number, such that $\mathcal{U}_{\beta,\pm N}$ has positive Hausdorff dimension if $\beta\in(1/(2N-1),\beta_c)$, and contains countably infinite many elements if $\beta\in(\beta_c,1/N)$. Moreover, there exists a second critical point $\alpha_c=\big[N+1-\sqrt{(N-1)(N+3)}\,\big]/2\in(1/(2N-1),\beta_c)$ such that $\mathcal{S}_{\beta,\pm N}$ has positive Hausdorff dimension if $\beta\in(1/(2N-1),\alpha_c)$, and contains countably infinite many elements if $\beta\in[\alpha_c,1/N)$.Comment: 23 pages, 4 figure