## Catastrophic Outbreak

I am enjoying Steven Strogatz second edition of “Non-linear Dynamics and Chaos”.

This post is motivated by this. The following gif relates to the model for insect outbreak. It shows the critical point parameter surface.

The differential equation: . Note the fixed point is unstable.

Not as nice and smooth as I’d like but fun nonetheless.

## The Grapes of Math

This is a very enjoyable read. The author has a very entertaining and engaging writing style. The book starts looks at psychological aspects of human relationship to numbers. Thereafter, we are taken on a journey through triangles, circles, conic sections, complex numbers, calculus, cellular automata and proof and more. Pi, e, and i are characters and the amazing appearance and connectedness is expressed. This is a very rich journey through history and exposition of the variety of applications some seemingly esoteric areas have found. I particularly enjoyed the sections on roulettes.

The book is not a technical explanation but invites the reader to play: whether it is origami to produce a parabola, rolling one coin over another, or looking for patterns in your cup to tea. However, the concepts are expressed clearly and convincingly.

The complex dynamic interaction between development of mathematics and its applications: motivations, personalities and serendipity in complex feedback loops , shows what a wonderfully human endeavour Mathematics is.

## On A Roll

This post is motivated by Bellos’ “Grapes of Math”:

Hyperboloid from straightlines

Train wheels:

I could not resist the “cardioid” roulette:

and just for fun rolling coins:

## Div, Grad, Curl, and All That: An Informal Text on Vector Calculus

This is an excellent informal introduction. It is written clearly, uses simple but illustrative worked examples with helpful graphics. It covers the title topics and uses electrostatics as a tangible application for illustration. There a lot of problems and solutions at the end.

## What to Expect

I found this question on Mathematica Stackexhange interesting. I learned a lot from wolfies answer. I post this not as an answer to the question of closed form but to illustrate consistency of a number of approaches.

The calculations take some time but it does not seem onerous.

Insights into the expectation can be obtained from the region of of interest.

ms[p_] :=

Show[Plot3D[(1 - x)^y, {x, 0, 1}, {y, 2, 5},

PlotStyle -> {LightBlue, Opacity[0.5]}

, PerformanceGoal -> "Speed", PlotRange -> {0, 1}],

Plot3D[(1 - x)^y, {x, 0, 1}, {y, 2, 5}, MeshFunctions -> {#3 &},

Mesh -> {{1 - p}}, MeshStyle -> {Red, Thick},

RegionFunction -> Function[{x, y, z}, (1 - x)^y "Speed", PlotRange -> {0, 1}],

AxesLabel -> (Style[#, 20] & /@ {x, y, (1 - x)^y}), ImageSize -> 500]

Manipulate[ms[t], {{t, 0, "p"}, 0, 1, Appearance -> "Labeled"}]

Or more easily with:

Manipulate[

RegionPlot[(1 - x)^y < 1 - p, {x, 0, 1}, {y, 2, 5}], {p, 0, 1}]

Now to numerically calculate the expectation:

int[p_] :=

Integrate[

Boole[{x, y} \[Element]

ImplicitRegion[0 < x < 1 && 2 < y < 5, {x, y}]] x/3, {x, 0,

1}, {y, Log[1 - p]/Log[1 - x], 5}]

td = TransformedDistribution[(1 - x)^y, {x, y} \[Distributed]

UniformDistribution[{{0, 1}, {2, 5}}]];

d[p_] := Probability[z < 1 - p, z \[Distributed] td]

result[p_] := int[p]/d[p]

ex[p_] :=

Expectation[

x \[Conditioned] (1 - x)^y < 1 - p, {x, y} \[Distributed]

UniformDistribution[{{0, 1}, {2, 5}}]]

Definition[rp]

rv[p_, n_] :=

Module[{r},

r = RandomVariate[UniformDistribution[{{0, 1}, {2, 5}}], n];

Mean[Pick[r, ((1 - #1)^#2 < 1 - p) & @@@ r][[All, 1]]]]

rv1000[p_] := rv[p, 1000];

The above firstly uses wolfies method, then just asks *Mathematica* to calculate explicitly and the simulates.

Tabulating:

grid = Prepend[

Through[{Identity, result, ex, rv1000}[#]] & /@

Range[0.1, 0.9, 0.1],

Style[#, Bold] & /@ {"p", "Derived", "Expectation",

"Estimated (N=1000)"}];

gr = Grid[grid, Frame -> All,

Background -> {None, {{White, LightBlue}}}]

Plotting (using `Table`

to minimize computation time):

plt = ListLinePlot[Table[ex[t], {t, 0.1, 0.9, 0.1}], Frame -> True,

FrameLabel -> {"p", "Conditional Expectation x"}, BaseStyle -> 16,

ImageSize -> 600]

This does little to answer the closed form question but I found it fun.

## How Not to Be Wrong

.

This is a wonderful book. I thoroughly enjoyed it. I learned Mathematical, historical and cultural jewels that I very much look forward to contemplating further or delving into. The author is refreshingly forthright in exploring the impediments to Mathematics Education and its appreciation in society at large. He does this from the position of a working Mathematician and educator. A broad array of concepts are covered and their exposition are an intricate weaving of historical events and individuals with current events and instructive toy examples. Wald, Fisher, Pascal, Shannon, Galton, Hilbert and many more appear at varying times. These insights into these individuals was remarkably more complex for the amount of words: strengths and weaknesses of their contributions (and the positions, e.g Galton and eugenics).

One of the greatest joys of this book is the discussions are neither superficial nor simplistic. The author starts from the simple and when you think you have digested what is at hand, he add another wrinke and another and enriches your understanding. He supports his case that Mathematics is the extension of common sense. Particularly informative were the discussions of the law of large numbers, lotteries, the concept of regression to the mean. The importance of the role of uncertaintuy in our lives, acknowledging its existence and the value of quantifying marries well with Nate Silvers “The Signal and The Noise” (Silver also discussed in the book).

The author discusses the “cult of genius” and the obstacle it creates to recruitment and retention of Mathematicians and the enrichment of other disciplines by Mathematics. This was compelling and struck me and derived from the insights of one who had negotiated the journey (though admittedly a prodigy) as well as tasked with guiding and supporting others through it.

There is a lot in this book. It is, however, more than content. It is a challenge to think more deeply about everyday life, about culture, about history and the seeing the “unreasonable effectiveness of Mathematics”.

## Money money

I am thoroughly enjoying Jordan Ellenberg’s “How not to be wrong”. This small post is some fun looking at Buffon’s coin puzzle. This shows a sample of twenty coins of varying radius relative to the underlying square. The fair game arises when the radius to length of square side is . The fraction in the boxes are the observed number of coins in the square and the predicted fraction. The value below the box is the ratio of radius to square length.