Patterns in rational base number systems

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the ad\'ele ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums

Oligomeric Status and Nucleotide Binding Properties of the Plastid ATP/ADP Transporter 1: Toward a Molecular Understanding of the Transport Mechanism

Background: Chloroplast ATP/ADP transporters are essential to energy homeostasis in plant cells. However, their molecular mechanism remains poorly understood, primarily due to the difficulty of producing and purifying functional recombinant forms of these transporters. Methodology/Principal Findings: In this work, we describe an expression and purification protocol providing good yields and efficient solubilization of NTT1 protein from Arabidopsis thaliana. By biochemical and biophysical analyses, we identified the best detergent for solubilization and purification of functional proteins, LAPAO. Purified NTT1 was found to accumulate as two independent pools of well folded, stable monomers and dimers. ATP and ADP binding properties were determined, and Pi, a co-substrate of ADP, was confirmed to be essential for nucleotide steady-state transport. Nucleotide binding studies and analysis of NTT1 mutants lead us to suggest the existence of two distinct and probably inter-dependent binding sites. Finally, fusion and deletion experiments demonstrated that the C-terminus of NTT1 is not essential for multimerization, but probably plays a regulatory role, controlling the nucleotide exchange rate. Conclusions/Significance: Taken together, these data provide a comprehensive molecular characterization of a chloroplas

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Basic properties of shift radix systems

Certain dynamical systems on the set of integer vectors are introduced and their basic properties are described. Applications to beta-expansions and canonical number systems reveal unexpected relations between different radix representation concepts

Stress, nutrition and parental care in a teleost fish

Parental care is an essential life-history component of reproduction for many animal species, which entails a suite of behavioural and physiological investments to enhance offspring survival. These investments can incur costs to the parent, reducing their energetic and physiological condition, future reproductive capabilities, and survival. In fishes, relatively few studies have focused on how these physiological costs are mediated. Male smallmouth bass provide parental care for developing offspring until the brood reaches independence. During this energetically demanding life stage, males cease active foraging as they vigorously defend their offspring. Experimental manipulation of cortisol levels (via implantation) and food (via supplemental feeding) in parental males was used to investigate the fitness consequences of parental care. Improving the nutritional condition of nest guarding males increased their reproductive success by reducing premature nest abandonment. However, supplemental feeding and cortisol treatment had no effect on parental care behaviours. Cortisol treatment reduced plasma lymphocyte numbers, but increased neutrophil and monocyte concentrations, indicating a shift in immune function. Supplemental feeding improved the physiological condition of parental fish by reducing the accumulation of oxidative injury. Specifically, supplemental feeding reduced the formation of 8-hydroxy-2'-deoxyguanosine (8-OHdG) on DNA nucleotides. Increasing the nutritional condition of parental fish can reduce the physiological cost associated with intensive parental activity and improve overall reproductive success, illustrating the importance of nutritional condition as a key modulator of parental fitness