Dynamical zeta functions and Kummer congruences

We establish a connection between the coefficients of Artin-Mazur zeta-functions and Kummer congruences. This allows to settle positively the question of the existence of a map T such that the number of fixed points of the n-th iterate of T is equal to the absolute value of the 2n-th Euler number. Also we solve a problem of Gabcke related to the coefficients of Riemann-Siegel formula.Comment: 12 pages, AMS-LaTe

Algorithms for determining integer complexity

We present three algorithms to compute the complexity $\Vert n\Vert$ of all natural numbers $n\le N$. The first of them is a brute force algorithm, computing all these complexities in time $O(N^2)$ and space $O(N\log^2 N)$. The main problem of this algorithm is the time needed for the computation. In 2008 there appeared three independent solutions to this problem: V. V. Srinivas and B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune [3]. All three are very similar. Only [11] gives an estimation of the performance of its algorithm, proving that the algorithm computes the complexities in time $O(N^{1+\beta})$, where $1+\beta =\log3/\log2\approx1.584963$. The other two algorithms, presented in [7] and [3], were very similar but both superior to the one in [11]. In Section 2 we present a version of these algorithms and in Section 4 it is shown that they run in time $O(N^\alpha)$ and space $O(N\log\log N)$. (Here $\alpha = 1.230175$). In Section 2 we present the algorithm of [7] and [3]. The main advantage of this algorithm with respect to that in [11] is the definition of kMax in Section 2.7. This explains the difference in performance from $O(N^{1+\beta})$ to $O(N^\alpha)$. In Section 3 we present a detailed description a space-improved algorithm of Fuller and in Section 5 we prove that it runs in time $O(N^\alpha)$ and space $O(N^{(1+\beta)/2}\log\log N)$, where $\alpha=1.230175$ and $(1+\beta)/2\approx0.792481$.Comment: 21 pages. v2: We improved the computations to get a better bound for $\alpha

Gaussian variables, polynomials and permanents

AbstractWe prove that given n unit vectors (aj)nj=1 in a complex Hilbert space there exists a unit vector x such that|〈x,a1〉…〈x,an〉|≱n−n2.This answers a question about the best constant in an inequality for polynomials in several variables. The proof depends on a relation between gaussian variables and permanents

Learner-generated digital media (LGDM) as an assessment tool in tertiary science education: A review of literature

© 2018, The International Academic Forum (IAFOR). All rights reserved. Learner-Generated Digital Media (LGDM) in tertiary science education focuses on research skills, inquiry, active learning, teamwork, and collaboration. LGDM across disciplines is under-theorised, under-researched, and only in its early development. This paper evaluates the research in the field of LGDM in tertiary science education. The literature review had four stages – identification, screening, filtering, and selection of relevant scholarly research. Results showed that research in the field of LGDM assignments had been done without a systematic approach to designing, implementing, and evaluating the assessment task. Most studies neglected student digital media training and are characterised by a lack of compelling marking rubrics or strategies to ensure efficient groupwork. Studies also lack rigorous methodologies for data capture to evaluate the intervention and they use small sample size cohorts and different digital media types that require different sets of production skills. With the empirical data available, validation of the benefits of LGDM assignments in science education is not possible, and studies have limited scalability. These gaps in the literature create a need to develop theoretical models for the design, implementation, and evaluation of LGDM in the classroom. This paper discusses future research needs in this field and the implications for assessment design