The arithmetic derivative and Leibniz-additive functions

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$f(mn)=f(m)h_f(n)+f(n)h_f(m)$$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions

Summative assessment of clinical practice of student nurses : a review of the literature

Objectives: To describe assessment of nursing student’s clinical practice concerned nursing education. Design: Systematic review and synthesis of qualitative and quantitative studies. Data sources: The data were collected with the support of an information specialist from scientific databases Cinahl, PubMed, Medic, ISI Web of Science, Cochrane library and Eric published in January 2000 – May 2014. All of the included studies citations were also performed. Methods: 725 articles concerned with nurse student clinical practice assessment were identified. After inclusion and exclusion criteria were met 23 articles for selected for critical review. Two independent reviewers selected the studies according to the inclusion criteria. These articles were analyzed using content analysis. Results: Findings suggest that the assessment process of nursing students’ clinical practice lacks consistency, it is open to the subjective bias of the assessor and the quality of assessment varies greatly. Nursing students clinical assessment was divided into 3 themes: acts (things to do) before final assessment, the actual final assessment situation and the acts after the final assessment situation. Mentors and students need orientation to the assessment process and to the paperwork by teachers. Terminology on evaluation forms is sometimes so difficult to grasp, that the mentors did not understand what they mean. There is no consensus about written assignments’ ability to describe the students’ skills. Mentors have timing problems to ensure relevant assessment of student nurses. At the final interview students normally self assess their performance, the mentor assesses by interview and by written assignments whether the student has achieved the criteria and role of the teacher is to support the mentor and the student in appropriate assessment. The variety of patient treatment environments in which nursing students do their clinical practice periods is challenging also for the assessment of nursing students’ expertise. Mentors alone want that clinical practice is a positive experience and it might lead to higher grades than what nurse student competency earns. It is very rare that students fail their clinical practice, if the student does not achieve the clinical competencies they are allowed to have extra time in clinical areas until they will be assessed as competent. Conclusions: This systematic review provides a description of challenges in nursing students’ assessment in clinical settings. Further research needs to be carried out to have more knowledge of final assessment in the end of the clinical practice. Through further research it will be possible to have better methods for high quality assessment processes and feedback to nurse students. Quality in assessment provides better nurses and therefore better patient safety

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

Let $pinmathbb P$ and $sinmathbb R$, and suppose that$emptysetne Psubsetmathbb P$ is finite.Given $ninmathbb Z_+$, let $n\u27$, $n\u27_p$, and $n\u27_P$ denote respectively its arithmetic derivative, arithmetic partial derivative with respect to~$p$,and arithmetic subderivative with respect to~$P$. We study the asymptotics of $$sum_{1le nle x}frac{n\u27}{n^s},,sum_{1le nle x}frac{n\u27_p}{n^s},quad{rm and},,sum_{1le nle x}frac{n\u27_P}{n^s}.$$ We also show that the abscissa of convergence of the corresponding Dirichlet series equals~two