Non-normality of continued fraction partial quotients modulo q

It is well known that almost all real numbers (in the sense of Lebesgue measure) are normal to base q where q ≥ 2 is any integer base

Graphene transistors are insensitive to pH changes in solution

We observe very small gate-voltage shifts in the transfer characteristic of as-prepared graphene field-effect transistors (GFETs) when the pH of the buffer is changed. This observation is in strong contrast to Si-based ion-sensitive FETs. The low gate-shift of a GFET can be further reduced if the graphene surface is covered with a hydrophobic fluorobenzene layer. If a thin Al-oxide layer is applied instead, the opposite happens. This suggests that clean graphene does not sense the chemical potential of protons. A GFET can therefore be used as a reference electrode in an aqueous electrolyte. Our finding sheds light on the large variety of pH-induced gate shifts that have been published for GFETs in the recent literature

Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate

Successions in Words and Compositions

We consider words over the alphabet [k] = {1, 2, . . ., k}, k ≥ 2. For a fixed nonnegative integer p, a p-succession in a word w 1w 2 . . . w n consists of two consecutive letters of the form (w i, w i + p), i = 1, 2, . . ., n-1. We analyze words with respect to a given number of contained p-successions. First we find the mean and variance of the number of p-successions. We then determine the distribution of the number of p-successions in words of length n as n (and possibly k) tends to infinity; a simple instance of a phase transition (Gaussian-Poisson-degenerate) is encountered. Finally, we also investigate successions in compositions of integers. © 2012 Springer Basel AG

Separation of the maxima in samples of geometric random variables

We consider samples of n geometric random variables ω 1 ω 2 · · · ω n where ({ω j = i}=pq i-1, for 1 ≤ j ≤ n, with p+q=1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice's method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random variables with at least two maxima