In this paper we study Lame´equations L n;B y ¼ 0 in so-called algebraic form, having only algebraic functions as solution. In particular we provide a complete list of all finite groups that occur as the monodromy groups, together with a list of examples of such equations. We show that the set of such Lame´equations with ne1=2 þ Z is countable, up to scaling of the equation. This result follows from the general statement that the set of equivalent secondorder equations, having algebraic solutions and all of whose integer local exponent differences are 1, is countable. r Second-order equations with finite monodromyAlthough ordinary second-order linear differential equations have been familiar in the literature for a few centuries now, there remain a number of interesting open questions which are still unsolved today. In this paper we shall be interested in second-order Fuchsian equations, i.e. ordinary linear equations having at most regular singularities at all points of the Riemann sphere P 1 (including N).

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We applied a queuing model to inform ventilator capacity planning during the first wave of the COVID-19 epidemic in the province of British Columbia (BC), Canada. The core of our framework is a multi-class Erlang loss model that represents ventilator use by both COVID-19 and non-COVID-19 patients. Input for the model includes COVID-19 case projections, and our analysis incorporates projections with different levels of transmission due to public health measures and social distancing. We incorporated data from the BC Intensive Care Unit Database to calibrate and validate the model. Using discrete event simulation, we projected ventilator access, including when capacity would be reached and how many patients would be unable to access a ventilator. Simulation results were compared with three numerical approximation methods, namely pointwise stationary approximation, modified offered load, and fixed point approximation. Using this comparison, we developed a hybrid optimization approach to efficiently identify required ventilator capacity to meet access targets. Model projections demonstrate that public health measures and social distancing potentially averted up to 50 deaths per day in BC, by ensuring that ventilator capacity was not reached during the first wave of COVID-19. Without these measures, an additional 173 ventilators would have been required to ensure that at least 95% of patients can access a ventilator immediately. Our model enables policy makers to estimate critical care utilization based on epidemic projections with different transmission levels, thereby providing a tool to quantify the interplay between public health measures, necessary critical care resources, and patient access indicators.

Criminal justice systems are complex. They are composed of several major subsystems, including the police, courts, and corrections, which are in turn composed of many minor subsystems. Predicting the response of a criminal justice system to change is often difficult. Mathematical modeling and computer simulation can serve as powerful tools for understanding and anticipating the behavior of a criminal justice system when something does change. The focus of this chapter is on three different approaches to modeling and simulating criminal justice systems: process modeling, discrete event simulation, and system dynamics. Recent advances in these modeling techniques combined with recent large increases in computing power make it an ideal time to explore their application to criminal justice systems. This chapter reviews these three approaches to modeling and simulation and presents examples of their application to the British Columbia criminal justice system in order to highlight their usefulness in exploring different types of “what-if” scenarios and policy proposals.

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