A cross-intersection theorem for subsets of a set

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of all subsets of $\{1, \dots, n\}$ of size at most $k$. We show that if $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $$|\mathcal{A}||\mathcal{B}| \leq \sum_{i=0}^r {m-1 \choose i-1} \sum_{j=0}^s {n-1 \choose j-1},$$ and equality holds if $\mathcal{A} = \{A \in {[m] \choose \leq r} \colon 1 \in A\}$ and $\mathcal{B} = \{B \in {[n] \choose \leq s} \colon 1 \in B\}$. Also, we generalise this to any number of such cross-intersecting families.Comment: 12 pages, submitted. arXiv admin note: text overlap with arXiv:1212.695

Factors determining career choice

The answers that people provide to these questions: why do people work? why do they decide on the fields which they choose? and what factors affect their decision? are very important to career planning and subsequent satisfaction during the adult years. After all, work affects most persons between the ages of 16 and 61, and the decisions adolescents make about their work, occupations, and careers will significantly affect their future social relationships and leisure-time activities. It is evident then that work is a major part of human experience. Many young people appear to sense that it is through work that they must ultimately validate their adult status and acquire a measure of power and self- determination. Work is so central to most of people's daily existence that their entire outlook is affected by it. In essence, a vocational decision implies a lifestyle decision. For choice of career is not an event which can be located at one point in time. It is a process which stretches back into childhood where basic personality characteristics begin to be formed {Stephens, 1970; Gothard, 1985 ). In making a choice, the individual will seek a career which s/he sees as desirable, as one in which s/he will have the best chance of realising the various needs, hopes and expectations which at the moment of choice s/he believes to be important.peer-reviewe

Developing a theme in social studies : an alternative approach

The Council of Europe in its documents on the place of History in Secondary schools, states that 'the teacher who stands in front of the class with a chalk is almost soundly condemned as the teacher who uses the stick'. The statement, directed to teachers who still insist on boring their students by chalking and talking, leaves no room for interpretation. It is direct and clear, emphasizing that: beyond the world of soar throats and chalky air there is a wealth of resources that can inject interest and curiosity, make possible variation of teaching methods, provide stimuli for the fantasy world of students, acquaint students with research techniques, make it easier for students to pursue self-instruction, show the relation of previously isolated facts, relate abstract generalisations with realistic details and, above all, add clarity and precision to the way the lesson is presented.peer-reviewe

Webcast courses in Medical Genetics and next generation sequencing

The European School of Genetic Medicine organised the 26th Course in Medical Genetics and the 2nd Course in Next Generation Sequencing, between the 12th and 20th May 2013. Both courses were webcast live from the Bologna University Residential Centre, Bertinoro, Italy. Participants in Malta attended these courses at the University’s Medical School. The course in Medical Genetics covered various aspects of this rapidly developing field of Medicine. The different methodologies used in human genome analysis, an introduction to Next Generation Sequencing (NGS), approaches to clinical and molecular genetics, complex genetic disorders, therapy and gene regulation, were covered. The second course provided a comprehensive insight into NGS technologies, from the basics to the new world of disease gene identification by hand-held devices. It also covered insights into bioinformatics challenges, sample preservation and trans-omic studies, and new frontiers including the investigation of single cells and of the non-coding genome.peer-reviewe

Teacher cognition in language teaching: A review of research on what language teachers think, know, believe, and do

This paper reviews a selection of research from the field of foreign and second language teaching into what is referred to here as teacher cognition – what teachers think, know, and believe and the relationships of these mental constructs to what teachers do in the language teaching classroom. Within a framework suggested by more general mainstream educational research on teacher cognition, language teacher cognition is here discussed with reference to three main themes: (1) cognition and prior language learning experience, (2) cognition and teacher education, and (3) cognition and classroom practice. In addition, the findings of studies into two specific curricular areas in language teaching which have been examined by teacher cognition – grammar teaching and literacy – are discussed. This review indicates that, while the study of teacher cognition has established itself on the research agenda in the field of language teaching and provided valuable insight into the mental lives of language teachers, a clear sense of unity is lacking in the work and there are several major issues in language teaching which have yet to be explored from the perspective of teacher cognition