Normality of different orders for Cantor series expansions

Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of all orders in $S$ but not $Q$-normal of all orders not in $S$ has full Hausdorff dimension. If the function $k \mapsto 1_S(k)$ is computable, then there exist computable examples. For example, there exists a computable basic sequence $Q$ where the set of numbers normal of all even orders and not normal of all odd orders has full Hausdorff dimension. This is in strong constrast to the $b$-ary expansions where any real number that is normal of order $k$ must also be normal of all orders between $1$ and $k-1$. Additionally, all numbers we construct satisfy the unusual condition that block frequencies sampled along non-trivial arithmetic progressions don't converge to the expected value. This is also in strong contrast to the case of the $b$-ary expansions, but more similar to the case of the continued fraction expansion. As a corollary, the set of $Q$-normal numbers that are not normal when sampled along any non-trivial arithmetic progression has full Hausdorff dimension.Comment: 24 page

On the Hausdorff dimension of some sets of numbers defined through the digits of their QQ-Cantor series expansions

Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we consider difference sets associated with various notions of normality and sets of numbers with a prescribed range of digits.Comment: 13 pages, 1 figur

Unexpected distribution phenomenon resulting from Cantor series expansions

We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones.Comment: 32 page

Children's interactions with interactive toy technology

Abstract Digital toys offer the opportunity to explore software scaffolding through tangible interfaces that are not bound to the desktop computer. This paper describes the empirical work completed by the CACHET (Computers and Children's Electronic Toys) project team investigating young children's use of interactive toy technology. The interactive toys in question are plush and cuddly cartoon characters with embedded sensors that can be squeezed to evoke spoken feedback from the toy. In addition to playing with the toy as it stands, the toy can be linked to a desktop PC with compatible software using a wireless radio connection. Once this connection is made the toy offers hints and tips to the children as they play with the accompanying software games. If the toy is absent, the same hints and tips are available through an on-screen animated icon of the toy's cartoon character. The toys as they stand are not impressive as collaborative learning partners, as their help repertoire is inadequate and even inappropriate. However, the technology has potential: children can master the multiple interfaces of toy and screen and, when the task requires it and the help provided is appropriate, they will both seek and use it. In particular, the cuddly interface experience can offer an advantage and the potential for fun interfaces that might address both the affective and the effective dimensions of learners' interactions

Normal equivalencies for eventually periodic basic sequences

W. M. Schmidt, A. D. Pollington, and F. Schweiger have studied when normality with respect to one expansion is equivalent to normality with respect to another expansion. Following in their footsteps, we show that when $Q$ is an eventually periodic basic sequence, that $Q$-normality and $Q$-distribution normality are equivalent to normality in base $b$ where $b$ is dependent on $Q$. We also show that boundedness of the basic sequence is not sufficient for this equivalence.Comment: 7 page

The politics of tourism development: a case of dual governance in Tobago

The purpose of this paper is to report on an exploratory study of the role that tourism policy plays in influencing tourism development. The research is based on two questions: How does tourism policy influence tourism development in the small twin-island state of Trinidad and Tobago? and How does an arrangement of dual governance in a small twin-island state promote tourism development? The exploration demonstrates that tourism development in the context of Tobago, the smaller of the two islands, has been slowed as a result of dual governance and hence dual policy arrangements

Talking about teaching in English: Swedish university lecturers’ experiences of changing teaching language

This study documents the experiences of Swedish university lecturers when they change from teaching in their first language to teaching in English. Eighteen lecturers from two Swedish universities took part in a training course for teachers who need to give content courses in English. As part of the course the participants gave mini-lectures in their first language in a subject area that they usually teach. The following week, the lecturers gave the same lectures again, this time in English. The pairs of lectures were videoed and commented on by the lecturers themselves and the whole course cohort in an online discussion forum (an input of approximately 60 000 words). In addition, twelve of the lecturers were interviewed about their experiences of changing language in this way (total of 4 hours of recorded material). The paper presents a qualitative analysis of the thoughts and experiences expressed by the lecturers in their online discussions and in the interviews concerning the process of changing the language of instruction to English. These results are presented as nine themes. Nine recommendations for teachers changing to teaching in English are also presented. The findings replicate those of earlier studies with one notable exception: the lecturers in this study were acutely aware of their limitations when teaching in English. It is suggested that this may be due to the lecturers’ relative inexperience of English-medium instruction

Superconductivity in heavily compensated Mg-doped InN

We report superconductivity in Mg-doped InN grown by molecular beam epitaxy. Superconductivity phase transition temperature occurs Tc = 3.97 K as determined by magnetoresistance and Hall resistance measurements. The two-dimensional (2D) carrier density of the measured sample is n2D = 9×1014 cm−2 corresponding to a three-dimensional (3D) electron density of n3D = 1.8×1019 cm−3 which is within the range of values between Mott transition and the superconductivity to metal transition. We propose a plausible mechanism to explain the existence of the superconductivity in terms of a uniform distribution of superconducting InN nanoparticles or nanosized indium dots forming microscopic Josephson junctions in the heavily compensated insulating bulk InN matrix