Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words

A word $u$ defined over an alphabet $\mathcal{A}$ is $c$-balanced ($c\in\mathbb{N}$) if for all pairs of factors $v$, $w$ of $u$ of the same length and for all letters $a\in\mathcal{A}$, the difference between the number of letters $a$ in $v$ and $w$ is less or equal to $c$. In this paper we consider a ternary alphabet $\mathcal{A}=\{L,S,M\}$ and a class of substitutions $\phi_p$ defined by $\phi_p(L)=L^pS$, $\phi_p(S)=M$, $\phi_p(M)=L^{p-1}S$ where $p>1$. We prove that the fixed point of $\phi_p$, formally written as $\phi_p^\infty(L)$, is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of $p$. We also show that both these bounds are optimal, i.e. they cannot be improved.Comment: 26 page

Threshold resonance and controlled filtering in quantum star graphs

We design two simple quantum devices applicable as an adjustable quantum spectral filter and as a flux controller. Their function is based upon the threshold resonance in a F\"ul\"op-Tsutsui type star graph with an external potential added on one of the lines. Adjustment of the potential strength directly controls the quantum flow from the input to the output line. This is the first example to date in which the quantum flow control is achieved by addition of an external field not on the channel itself, but on other lines connected to the channel at a vertex.Comment: ReVTeX format, 4 pages, 6 figure

Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers

We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $\beta$-fractional positions, which may arise as a result of addition of two $\beta$-integers. For the infinite word $u_\beta$ coding distances between consecutive $\beta$-integers, we determine precisely also the balance. The word $u_\beta$ is the fixed point of the morphism $A \to A^{m-1}B$ and $B\to A^{m-n-1}B$. In the case $n=1$ the corresponding infinite word $u_\beta$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $\beta$-integers are related.Comment: 15 page

Quantum anholonomy with topology change

We study a family of closed quantum graphs described by one singular vertex of order n=4. By suitable choice of the parameters specifying the singular vertex, we can construct a closed sequence of paths in the parameter space that physically corresponds to the smooth interpolation of different topologies - a ring, separate two lines, separate two rings, two rings with a contact point. We find that the spectrum of a quantum particle on this family of graphs shows the quantum anholonomy.Comment: 6 pages, 10 figure