Uniqueness of a convex sum of products of projectors

Relative to a given factoring of the Hilbert space, the decomposition of an operator into a convex sum of products over sets of distinct 1-projectors, one set linearly independent, is unique.Comment: 4 pages. v2: Minor clarifications in Section III; as accepted for publication in J Math Phy

How students blend conceptual and formal mathematical reasoning in solving physics problems

Current conceptions of expert problem solving depict physical/conceptual reasoning and formal mathematical reasoning as separate steps: a good problem solver first translates a physical Current conceptions of quantitative problem-solving expertise in physics incorporate conceptual reasoning in two ways: for selecting relevant equations (before manipulating them), and for checking whether a given quantitative solution is reasonable (after manipulating the equations). We make the case that problem-solving expertise should include opportunistically blending conceptual and formal mathematical reasoning even while manipulating equations. We present analysis of interviews with two students, Alex and Pat. Interviewed students were asked to explain a particular equation and solve a problem using that equation. Alex used and described the equation as a computational tool. By contrast, Pat found a shortcut to solve the problem. His shortcut blended mathematical operations with conceptual reasoning about physical processes, reflecting a view - expressed earlier in his explanation of the equation - that equations can express an overarching conceptual meaning. Using case studies of Alex and Pat, we argue that this opportunistic blending of conceptual and formal mathematical reasoning (i) is a part of problem-solving expertise, (ii) can be described in terms of cognitive elements called symbolic forms (Sherin, 2001), and (iii) is a feasible instructional target.Comment: Pre-reviewed draft, now published in Science Educatio

Kajian Potensi dan Rekomendasi Desa Lhokrukam Berbasis Desa Wisata, sebagai Alternatif Pembangunan Kota Tapaktuan

Kabupaten Aceh selatan dilihat dari potensi alamnya, sektor priwisata sangat produktif untuk dikembangkan, karena didukung oleh letak geografisnya. Tapaktuan sangat dikenal dengan objek wisata tapak kaki tuan tapa yang juga merupakan simbol dari Kabupaten Aceh Selatan. Bukan hanya objek wisata tersebut, banyak objek wisata lainnya yang ada di Tapaktuan menjadi objek wisata andalan salah satunya di desa Lhokreukam yang memiliki potensi wisata yang tinggi. Namun sayangnya pengembangan pariwisata di desa tersebut belum optimal. Oleh sebab itu penelitian ini bertujuan untuk memberikan strategi pengembangan desa lhokreukam menjadi desa wisata sebagai salah satu destinasi wisata baru di Aceh Selatan. Adapaun metode yang di gunakan adalah analisis SWOT. Hasil dari penelitian ini adalah adanya rekomendasi strategi perencanaan pengembangan desa lhokreukam menjadi desa wisata, sehingga semua potensi wisata yang ada di desa tersebut dapat di kembangkan secara optimal dan berkelanjutan, dan menjadi alternatif pilihan berwisata yang ada di kabupaten Aceh Selatan bagi wisatawan