Category Archives: Mathematics

Now Is NOT the Time

In response to the question:

What’s the most disturbing truth about raising children?

Heavens. Where to start?

I’ll limit my answer to 3 unavoidable facts:

  1. The children you have right now (2015), and any yet to be born, are going to have to deal with the collapse of the biosphere. That’s not hyperbole: that’s the consensus of roughly 14,000 climate scientists, worldwide.                 BioShock                                                                                                                   They will grow up and try to thrive in a world where the norm will be mass migrations, food and water shortages, spread of deadly disease (malaria and West Nile virus, to name but two), and endless, endless wildfires, way too many to keep under control.                                                                                                                                They will live with storms that have the power to level cities. And it will be  hotter, much hotter.

  2. They will also have to deal with a world where there will be many, many applicants for fewer and fewer jobs.                                                                                                                                                             We only need so many people to grow our food, build and repair our highways, sell us our Starbucks, even trade our stocks for us. We put 250,000 new people on the planet (over and above the death rate) Every. Single. Day.                                  BirthRate Even if only half make it to adulthood, that’s still about 46 Million people that will need jobs in about 15 years. And then 46 million newer, new people the year after that. And the year after that.
    I’ve seen lines like this for part-time work at Chipotle’s. Seriously.

  3. Unless you are a Tiger Mom (or Dad), or can afford a good private school ($11,000/yr), your children are going to grow up — no way to say this but to say it — very dumb indeed.

The US ranks 28th in the world, just above tiny Portugal, in the education of its populace (Singapore ranks #1).

Horrid for a country that still likes to think of itself as a ‘superpower.’

I had to explain to a Millennial just yesterday why Benjamin Franklin was a Big Deal.

In sum: You are raising children

  • who will live in air-conditioning maybe 8 hours a day, before the power grids have their mandatory daily brownouts;

  • who will probably work part-time at a poorly-paid franchise operation, and

  • who will be unfamiliar with basic history, geography, or mathematics.

Further, engrossing, accurate reading: 

A Special Moment in History (Bill McKibben)


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Of Honeybees and Helicopters

A trend that I’ve always found quite galling is the bantering about of “well-known facts” that are, in fact, complete myths — especially those that  skitter up against science for a moment before careening off into Cloud-Cuckoo Land.
Case in point: this particularly treacly bit of nonsense, intended as a Motivational Thought:

Gaaah! What nonsense!
Gaaah! What nonsense!

By the way, yes — it’s that Mary Kay, the MLM cosmetics queen who left behind a personal fortune of nearly $98 million, and whose personal motto was “God first, family second, career third.” But that is another rant for another day

Here is the real story, according to Cecil Adams at The Straight Dope:

“According to an account at physics.org, the story was initially circulated in German technical universities in the 1930s. Supposedly during dinner a biologist asked an aerodynamics expert about insect flight.The aerodynamicist did a few calculations and found that, according to the accepted theory of the day, bumblebees didn’t generate enough lift to fly.

“Hummeln nicht fliegen! Mein Gott!!!”

The biologist, delighted to have a chance to show up those arrogant SOBs in the hard sciences, promptly spread the story far and wide.

“Once he sobered up, however, the aerodynamicist surely realized what the problem was — a faulty analogy between bees and conventional fixed-wing aircraft. Bees’ wings are small relative to their bodies. If an airplane were built the same way, it’d never get off the ground. But bees aren’t like airplanes, they’re like helicopters. 

Kind of like this, but not really.
Kind of like this, but not really.

Their wings work on the same principle as helicopter blades — to be precise, ‘reverse-pitch semirotary helicopter blades,’ to quote one authority.A moving airfoil, whether it’s a helicopter blade or a bee wing, generates a lot more lift than a stationary one.”

The take-home lesson here is that there can be quite a difference between a real-life concept and its mathematical model — especially if the initial model doesn’t reflect the structural reality.

And now for your viewing pleasure, some really, really cute bees:

Hannah’s Sweets: My Thoughts on the Solution

In June of this year there was an outcry amongst UK sixteen-year-olds in the wake of this GCSE probability problem:

Hannah

So, speaking as a mathematician (and a former college professor), here’s my take on this:

First and foremost, you can be sure of two things:

  • There is a solution, and

  • It’s simple.

Test-makers at the public-school level do not have time to plow through great, long, unwieldy answers to basic maths problems.

We *really* don't want to see this.
We *really* don’t want to see this.

You may be caught off-guard momentarily, but calm down, use what you know , and Keep It Simple.

So, what do we know? Well …

The first thing that jumps out at me is that quadratic.

It would be easy to write in an equivalent form:

n2 – n – 90 = 0

becomes

n2 – n = 90

What number, when subtracted from its square, equals 90?

Why, 10 of course!

So now, “Show n2 – n – 90 = 0″ becomes the much simpler equivalent statement


“Show n = 10.”

sweets
Sweet, eh?

Note: we are talking about a whole number of candies, so a small, simple number like 10 makes sense.


Onwards: back to What You Know.

Pulling one out of 6 orange sweets,and then one of 5 (remaining) orange sweets, from a bag of n (10, remember?) sweets looks like this:

FracsA

This resolves to    FracsB

Recall that the probability is one- third, and you get

FracsD

For n = 10 you get the true statement
FracsE

QED.

I know it’s easy to armchair-quarterback these things when you’re not sweating in a timed test– for what it’s worth, that’s my approach.

The Physics of Vertical Takeoff

No, this isn’t trick photography. This is a very good pilot at this year’s Farnborough Air Show, performing a vertical takeoff in Boeing’s new Dreamliner — roughly 225,000 kg taking off like a rocket.

But — isn’t it common knowledge that

vertical flight = stalls = crash?

Remember Colgan Air 3407?

So — what’s going on here?

What you’re seeing is a vertical flight path. Flying horizontally first, the airplane pitches up until the nose is pointing straight into the sky.

Rocket

You don’t need thrust for this. Even gliders can do it. What you’re seeing is  kinetic energy (speed of the plane on the runway)  converted to potential energy —  and accelerating 250 tons to nearly 200 miles per hour builds tremendous kinetic energy! With so much potential energy, vertical flight can be maintained for several seconds, until the aircraft runs out of speed and stops in midair.

In the video, I count about 6 seconds of vertical flight before the pilot drops the nose. 

In aerobatics, this maneuver is called a stall turn or a hammerhead stall.

hammerhead

Numbers: 

For a craft weighing x kg you need g*x Newtons of thrust, minimum, for sustained vertical flight. For each metric ton of weight you need around 9.81 kN of thrust. The Dreamliner has a operational empty weight of 225 tons, so it would need 2453 kN of thrust to sustain a vertical climb. Its 2 GEnx engines, each producing 330 kN, don’t provide nearly enough power. This is why it is necessary to pitch up to vertical while shedding speed — making this awe-inspiring manouever  possible without thrust.

Sunday Fun: Math, and 45 Shopping Days Left …

A little something to work the grey cells:
“The ARML (American Regions Math League) Power Contest is truly a unique competition in which a team of students is judged on its ability to discover a pattern, express the pattern in precise mathematical language, and provide a logical proof of its conjectures. Just as a team of students can be self-directed to solve each problem set, a teacher, math team coach, or math circle leader could take these ideas and questions and lead students into problem solving and mathematical discovery.

This book contains thirty-seven interesting and engaging problem sets presented at ARML Power Contests from 1994 to 2013. They are generally extensions of the high school mathematics classroom and often connect two remote areas of mathematics. Additionally, they provide meaningful problem situations for both the novice and the veteran mathlete.”

ARML

And now for something completely different ..
“The Redneck Plunger” — a featured item in The Lakeside Collection® catalog. I wonder about the people who think this is a fun Christmas gift …

37740_zm

Warsaw Days 5 and 6: Now, where were we?

So —

Back two days. After my successful plenary lecture (“Linear, Barotropic Equations: Existence, Uniqueness, and Properties of the Solution”), we celebrated by spending the evening exploring Łazienki Park , which is beautiful beyond description (but here goes): 188 acres, lakes, castles, peacocks, classical sculpture gardens. Golden-green, lush, cool, quiet. Lamplighters. I loved the lamplighters:

Lamplighter in Łazienki Park, August 2013

The park is still lit by gas lamps, which give off a completely different kind of light. Tiny step into the past, as you walk in the evening glow. Modern lamps are brighter, certainly, but also much harsher.

When our eyes couldn’t take in any more beauty, we headed to the top of a high promontory, where we had exquisite food at Qchnia Artystyczna , accompanied by the inevitable (and very good) 0.5 L of Tyskie , and all of Warsaw below us, twinkling in the dark.

Warsaw by Night

Note to readers: if in Poland, at a restaurant, and there’s something on the menu that has mushrooms — EAT IT. Poles are insane about mushrooms, and they’re not those horrid button things that we have to endure in the States. They’re what we would call artisanal. Carefully grown, sometimes foraged, so many varieties I have lost count. Heaven.

Oh, for my Math brothers and sisters, this was Wednesday’s lineup:

Wednesday, 14 August
9:30 – 10:20 A. Ioffe
Metric regularity in variational analysis
10:20 – 10:40 Coffee break
10:40 – 11:30 R. Guenther and C. C. Buchanan (That’s Me! My Math Author Name)
Linear Barotropic equations. Properties of the solutions
11:30 – 15:00 Lunch break
15:00 – 15:40 M. Frigon
On a notion of category depending on a functional
15:40 – 16:00 Coffee break
16:00 – 16:40 J. Pejsachowicz
Topology and bifurcation
16:40 – 17:00 Break
17:00 – 17:40 M. Koenig
S. Bernstein’s idea for bounding the gradient of solutions to the quasilinear Dirichlet’s problem

Well, as usual, it’s very late and I have an early morning tea with a new project director / co-researcher, a very respected, brilliant fellow in his field. That’s on top of my current writeup for publication, my book, looking for a new home for the PhD, and life in general. Got to sleep. More tomorrow.