A cone separation theorem

The notion of separation is extended here to include separation by a cone. It is shown that two closed cones, one of them acute and convex, can be strictly separated by a convex cone, if they have no point in common. As a matter of fact, an infinite number of convex closed acute cones can be constructed so that each of them is a separating cone.The research was done while the author was a visiting professor at the University of British Columbia, Vancouver, British Columbia, Canada.     

Differentiability of convex functions and the convex point of continuity property in Banach spaces

We show that ifX is a separable Banach space, then every continuous, convex, Gateaux differentiable function onX is Fréchet differentiable on a dense set if and only ifX* has theweak*-Convex Point of Continuity Property (C*PCP). Research completed while a visitor at the University of Alberta. Research supported in part by an H. R. MacMillan Fellowship from the University of British Columbia. Research partially supported by NSERC (Canada).     

The Quantity and Quality of Biology Laboratory Work in British Columbia High Schools

The purpose of this research was to survey the quantity and quality of laboratory work conducted by 11th- and 12th-grade biology students in British Columbia high schools to analyze student performance on laboratory-based questions on provincial examinations, and to examine the curriculum for recommended laboratory work. A pilot survey was used to produce a valid survey instrument that identified teachers' use of laboratory activities. Examination and curriculum data were extracted from Ministry of Education documents. The frequency of laboratory work was low, with quantitative activities much less frequent than qualitative. Laboratory exercises were confirmatory rather than investigative. Critical thinking and hypotheses formulation were seldom expected of the students. Many teachers reported that they infrequently or never used laboratory simulations or computer-based data collection. Although laboratories were equipped adequately, teachers claimed that the curriculum and provincial examination limited the scope for an inquiry-based course. Provincial examination results indicated that the frequency and type of laboratory work had no effect on scores on lab-based questions. Analysis of the laboratory activities recommended in the syllabus revealed that fewer laboratory activities were being conducted than are recommended in the curriculum guide outline. This study indicates that while the current philosophy of many biology teachers supports an investigative learning experience, little change in classroom practice has occurred in British Columbia schools.     

One dimensional stochastic partial differential equations and the branching measure diffusion

Summary An existence theorem in the spirit of Keisler [Ke], is proved for the simple one-dimensional diffusion equation driven by white noise modulated by a non-linear function of the solution. This is used to obtain a density and a Stochastic Partial Differential Equation in one dimension for the critical branching diffusion studied by Dawson et al.This research was carried out while the author was at the Department of Mathematics, University of British Columbia     

On multiplicative excessive functions of a branching Brownian motion

Summary In this note stochastic calculus is used to characterise multiplicative excessive functions of a binary branching Brownian motion with a constant creation rate. Some properties of the martingales given by invariant functions are studied. In particular, it is seen that these positive and unbounded martingales tend a.s. to 0 and are not square integrable. Informally speaking, they exhibit a clustering phenomenon in the underlying supercritical branching Brownian motion.This work was done while the author was visiting the University of British Columbia, Mathematical Department, and was partly supported by a NSERC grant. AMS 1980 subject classifications: primary 60J80, 60J65 secondary 60J60     

Peer interactions in a computer lab: reflections on results of a case study involving web-based dynamic geometry sketches

A case study, originally set up to identify and describe some benefits and limitations of using dynamic web-based geometry sketches, provided an opportunity to examine peer interactions in a lab. Since classes were held in a computer lab, teachers and pairs faced the challenges of working and communicating in a lab environment.Research has shown that particular teacher interventions provide motivation for the consideration of new ideas, and help uncover misunderstandings that may interfere with student progress [Towers, J. (1999). In what ways do teachers interventions interact with and occasion the growth of students’ mathematical understanding. Doctoral Dissertation, University of British Columbia, Unpublished]. Examples of student discourse presented here suggest that certain peer interactions act in similar ways—helping propel students towards new understanding. On the other hand, they also show that some peer interactions, although superficially similar to teacher interventions, may hamper student progress.     

Feller Property and Infinitesimal Generator of the Exploration Process

We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale. The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.     

Queuing network models for panel sizing in oncology

Motivated by practices and issues at the British Columbia Cancer Agency (BCCA), we develop queuing network models to determine the appropriate number of patients to be managed by a single physician. This is often referred to as a physician’s panel size. The key features that distinguish our study of oncology practices from other panel size models are high patient turnover rates, multiple patient and appointment types, and follow-up care. The paper develops stationary and non-stationary queuing network models corresponding to stabilized and developing practices, respectively. These models are used to determine new patient arrival rates that ensure practices operate within certain performance thresholds. Data from the BCCA are used to calibrate and illustrate the implications of these models.     

Optimization of the production of hydroelectric power systems with a variable head

We present in this paper a new approach for optimization of the production of hydroelectric power systems with a variable head. The problem is formulated as a minimum norm problem.The method takes into account the water head variation. To avoid underestimation of production for rising water levels and overestimation for falling water level, an average head (of begin and end of time step) is used. The method also takes into account the stochasticity of the river flows. Numerical results for a real system in operation including up to six reservoirs are reported for different water conditions. The proposed method is computationally efficient compared to other approaches.This work was supported by the National Research Council of Canada, Grant No. A-4146. The authors would like to acknowledge the data obtained from British Columbia Hydro.     

Observables and critical behaviour in fermionic matrix models

We review the properties of adjoint fermion one-, two-, and generic D-dimensional matrix models at large N. We derive and solve the complete sets of loop equations for the correlators of these models and examine the ensuing critical behavior. The topological 1/N-expansions are also constructed, and we discuss the applications of these matrix models to string theory and induced gauge theories.Work supported in part by the Natural Sciences and Engineering Research Council of Canada.Work supported in part by a University of British Columbia Graduate Fellowship.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 78–122, July, 1995.