rlopez

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12 years, 319 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

MaplePrimes Activity


These are Posts that have been published by rlopez

On 5/July/2017, Kitonum responded to the 3/July/2017 MaplePrimes question "How to perform double integration over subdomain" by providing code for a procedure IntOverDomain that implements Green's theorem applied to a planar region whose boundary is a simple, closed, rectifiable, oriented curve (SCROC by some authors).

I was intrigued. First, this is a significant extension of existing Maple functionalities. Second, the implementation admits boundaries defined piecewise with sections defined parametrically; or sections that are polygonal lines defined by a list of nodes.

But how was the line integral around such boundaries coded? In the worksheet "IntOverDomain_Deconstructed," I summarize the existing Maple functionality for implementing iterated double integrals over specified domains, then analyze how Kitonum coded Green's theorem as an extension of Maple's capabilities. After recognizing the great coding skills of Kitonum, I conclude with a short wishlist of related extensions that I would like to see added to Maple in the future.

 

Download the worksheet: IntOverDomain_Deconstructed.mw

While preparing for a recent webinar, I ran across something that didn't behave the same way in Maple 2017 as it did in previous releases. In particular, it was the failure of the over-dot notation for t-derivatives to display with the over-dot. Turns out that this is due to a change in behavior of typesetting that was detailed in the What's New page for Maple 2017, a page I had looked at many times in the last few months, but apparently didn't comprehend fully. The details are below.

Prior to Maple 2017, under the aegis of extended typesetting, the following two lines of code would alert Maple that the over-dot notation for t-derivatives should be used in the output display.

However, this changed in Maple 2017. Extended typesetting is now the default, but these two lines of code are no longer sufficient to induce Maple to display the over-dot in output. Indeed, we would now have

as output. The change is documented in the following paragraph

lifted from the help page 

Thus, it now takes the additional command

to induce Maple to display the over-dot notation in output.

I must confess that, even though I pored over the "What's New" pages for Maple 2017, I completely missed the import of this change to typesetting. I stumbled over the issue while preparing for an upcoming webinar, and frantically sent out help calls to the developers back in the building. Fortunately, I was quickly set straight on the matter, but was disappointed in my own reading of all the implications of the typesetting changes in Maple 2017. So perhaps this note will alert other users to the changes, and to the help page wherein one finds those essential bits of information needed to complete the tasks we set for ourselves.

And one more thing - I was cautioned that the "= true" was essential. Without it, the command would act as a query, echoing the present state of the setting, and not making the desired change to the setting.
 

Recently, I came across an addendum to a problem that appears in many calculus texts, an addendum I had never explored. It intrigued me, and I hope it will capture your attention too.

The problem is that of girding the equator of the earth with a belt, then extending by one unit (here, taken as the foot) the radius of the circle so formed. The question is by how much does the circumference of the belt increase. This problem usually appears in the section of the calculus text dealing with linear approximations by the differential. It turns out that the circumference of the enlarged band is 2*Pi ft greater than the original band.

(An alternate version of this has the circumference of the band increased by one foot, with the radius then being increased by 0.16 ft.)

The addendum to the problem then asked how high would the enlarged band be over the surface of the earth if it were lifted at one point and drawn as tight as possible around the equator. At first, I didn't know what to think. Would the height be some surprisingly large number? And how would one go about calculating this height.

It turns out that the enlarged and lifted band would be some 616.67 feet above the surface of the earth! This is significantly larger than the increase in the diameter of the original band. So, the result is a surprise, at least to me.

This is the kind of amusement that retirement affords. I heartily recommend both the amusement and the retirement. The supporting calculations can be found in the attached worksheet: Girding.mw

At 3:00 PM EST on Thursday, December 15, Maplesoft hosted a momentous hour in my life, my "retirement party" ending my career at Maplesoft. It was a day I had planned some four years ago when I dropped to a lighter schedule, and a day my wife has been awaiting for six years.

Jim Cooper, CEO at Maplesoft, presented a very brief sketch of some milestones in my life, including my high school graduation in 1958, BA in 1963, MS in 1966, PhD in 1970, jobs at the University of Nebraska-Lincoln, Memorial University of Newfoundland, and the Rose-Hulman Institute of Technology. There was a picture of me taken from my high school graduation yearbook. There was a cake. There were kind words about my contributions to Maple, including "Clickable Calculus," the term and its meaning.

I was handed the microphone - I knew what I wanted to say. My wife was present in the gathering. I pointed to her and said that all the congratulations should go to her who had waited so patiently for my retirement for six years. I thanked Maplesoft and all its employees for nearly 14 of the best years of my life, for I have thoroughly enjoyed my return to Canada and my work (more like play) at Maplesoft. 

It's been a great opportunity to be part of the Maple experience, and now it's time for new ones. There'll be more woodworking in my basement woodshop where I make mostly noise and sawdust, some extra travel, more exercise and fresh air, long-delayed household projects, and whatever else my mate of 49 years asks.

But the best part of all is that I'll still have a connection to Maplesoft - I'll continue doing two webinars a month, will maintain and update much of the content I've created for Maple while at Maplesoft, and contribute additional content of relevance to the Maple community. 

A population p(t) governed by the logistic equation with a constant rate of harvesting satisfies the initial value problem diff(p(t), t) = (2/5)*p(t)*(1-(1/100)*p(t))-h, p(0) = a. This model is typically analyzed by setting the derivative equal to zero and finding the two equilibrium solutions p = 50+`&+-`(5*sqrt(100-10*h)). A sketch of solutions p(t) for different values of a suggests that the larger equilibrium is stable; the smaller, unstable.

 

When a is less that the unstable equilibrium, p(t) becomes zero at a time t[e], and the population becomes extinct. If p(t) is not interpreted as pertaining to a population, its graph exists beyond t[e], and actually has a vertical asymptote between the two branches of its graph.

 

In the worksheet "Logistic Model with Harvesting", two questions are investigated, namely,

 

  1. How does the location of this vertical asymptote depend on on a and h?
  2. How does the extinction time t[e], the time at which p(t) = 0, depend on a and h?

To answer the second question, an explicit solution p = p(a, h, t), readily provided by Maple, is set equal to zero and solved for t[e] = t[e](a, h). It turns out to be difficult both to graph the surface t[e](a, h) and to obtain a contour map of the level sets of this function. Instead, we solve for a = a(t[e], h) and obtain a graph of a(h) with t[e] as a slider-controlled parameter.

 

To answer the first question, the explicit solution, which has the form alpha*tan(phi(a, h, t))*beta(h)+50, exhibits its vertical asymptote when phi(a, h, t) = -(1/2)*Pi. Solving this equation for t[a] = t[a](a, h) gives the time at which the vertical asymptote is located, a function that is as difficult to graph as t[e]. Again the remedy is to solve for, and graph, a = a(h), with t[a] as a slider-controlled parameter.

 

Download the worksheet: Logistic_with_Harvesting.mw

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