Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group \vG, we first construct vector spaces over \GF(p), $p$ a prime, by factorising \vG over appropriate normal subgroups. Then, by expressing \GF(p) in terms of the commutator subgroup of \vG, we construct alternating bilinear forms, which reflect whether or not two elements of \vG commute. Restricting to $p=2$, we search for ``refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of \vG is $\leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation'' of several distinct elements of \vG. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.Comment: 20 pages, 6 figures, 1 table; Version 2 - slightly polished, updated references; Version 3 - published version in SIGM

Authentic Identity: The Essence of How Successful “Ecopreneurs” Communicate

Entrepreneurs can no longer ignore the environmental consequences of their industrial practices. Those that do will most certainly fail. Ms. Schauch‘s research concentrates on the five distinguishing traits of a successful ―ecopreneur‖: (1) grounded by values and living by values; (2) belief in cause and passion for cause (3) resolute in mission and consistent across mission; (4) like-minded relationships and transparent relationships; and (5) learning for life and returning for life

Old School Catalog 1910-11, Chicago College of Dental Surgery

https://scholar.valpo.edu/oldschoolcatalogs/1017/thumbnail.jp

Prison(er) auto/biography, 'true crime', and teaching, learning, and research in criminology

The main aim of this essay is to explore prisoner life writing within the specific, richly and multiply dependent context of teaching and learning undergraduate criminology at an English university, from the authorial viewpoint of a teacher and her students as budding criminologists and co-authors. This article seeks to redress a continuing resistance to life history approaches in the teaching of criminology, despite the discipline being formally devoted to the understanding of the meaning and experience of imprisonment in all its forms and consequences. What follows is a trucated narrative of what students had to say on the fascinating subjects of prisoner auto/biography and its place in popular and expert discourses on crime, criminality, and punishment, contextualised within the academic discipline of criminology

Old School Catalog 1905-06, Chicago College of Dental Surgery

https://scholar.valpo.edu/oldschoolcatalogs/1033/thumbnail.jp

Old School Catalog 1907-08, Chicago College of Dental Surgery

https://scholar.valpo.edu/oldschoolcatalogs/1026/thumbnail.jp