We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By c...We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #<em>P</em> hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as <em>graph embeddings</em>, <em>discrete Morse theory </em>and<em> graph clustering</em>.展开更多
This paper is about a reflective practice happened in the author' s College English teaching.Since 2005,a 5-minute oral-presentation was required for all the students in turn,in order to raise opportunities for th...This paper is about a reflective practice happened in the author' s College English teaching.Since 2005,a 5-minute oral-presentation was required for all the students in turn,in order to raise opportunities for them to practice oral English.But the students' responses were always disappointing since they only took the task as a burden.After the teacher lift the limits on presentation time,form,sequence,partner,etc.,better learning outcomes were achieved.When reflecting on this change,Biggs'3P Models of "constructive alignment" is taken into account.Any teaching and learning activity (TLA) cannot put the students' and the teacher' s presage aside since the students' ability,interests,motivation as well as the teacher' s objective,assessment approach and so on are co-determinants of a successful TLA.Further suggestions are given based on this reflection at the end of the paper.展开更多
文摘We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #<em>P</em> hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as <em>graph embeddings</em>, <em>discrete Morse theory </em>and<em> graph clustering</em>.
文摘This paper is about a reflective practice happened in the author' s College English teaching.Since 2005,a 5-minute oral-presentation was required for all the students in turn,in order to raise opportunities for them to practice oral English.But the students' responses were always disappointing since they only took the task as a burden.After the teacher lift the limits on presentation time,form,sequence,partner,etc.,better learning outcomes were achieved.When reflecting on this change,Biggs'3P Models of "constructive alignment" is taken into account.Any teaching and learning activity (TLA) cannot put the students' and the teacher' s presage aside since the students' ability,interests,motivation as well as the teacher' s objective,assessment approach and so on are co-determinants of a successful TLA.Further suggestions are given based on this reflection at the end of the paper.
