BIOE 402. Medical Technology Assessment. 2 or 3 hours. Bioentrepreneur course. Assessment of medical technology in the context of commercialization. Objectives, competition, market share, funding, pricing, manufacturing, growth, and intellectual property; many issues unique to biomedical products. Course Information: 2 undergraduate hours. 3 graduate hours. Prerequisite(s): Junior standing or above and consent of the instructor.

“Bioinformatics” 특집을 내면서

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Distance-Regular Graphs

Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

We discuss representations of the projective line over a ringR with 1 in a projective space over some (not necessarily commutative) fieldK. Such a representation is based upon a (K, R)-bimoduleU. The points of the projective line overR are represented by certain subspaces of the projective space ℙ(K, U ×U) that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.

Projective representations i. projective lines over rings

Recently, a number of interesting relations have been discovered between genera- lised 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 G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups. Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G 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 G is 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 G. 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.

/pdf/factor-group-generated-polar-spaces-and-multi-qudits-3e87h9mi6f.pdf

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

https://www.geometrie.tuwien.ac.at/havlicek/pub/complement.pdf

On bijections that preserve complementarity of subspaces

As a continuation of our previous work (Havlicek and Saniga 2007 J. Phys. A: Math. Gen. 40 F943–52 (Preprint 0708.4333)) an algebraic geometrical study of a single d-dimensional qudit is made, with d being any positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring . Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of . We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes. The operators are also seen to be structured into disjoint 'layers' according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.

/pdf/projective-ring-line-of-an-arbitrary-single-qudit-1375ujk094.pdf

Projective ring line of an arbitrary single qudit

We discuss representations of the projective line over a ring $R$ with 1 in a projective space over some (not necessarily commutative) field $K$. Such a representation is based upon a $(K,R)$-bimodule $U$. The points of the projective line over $R$ are represented by certain subspaces of the projective space $P(K,U\times U)$ that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.

Hans Havlicek

Papers

Projective representations i. projective lines over rings

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

On bijections that preserve complementarity of subspaces

Projective ring line of an arbitrary single qudit

Projective Representations I. Projective lines over rings