# Let ε < 0.

## 09.19.09

### Oh! The places waves go!

Filed under: Harmonic analysis, Science humor — Travis @

–Kate Carlisle, Haverford College

Modeled on “Oh! The Places You’ll Go,” by Dr. Seuss.

Congratulations!
You’re off to learn many

You have math in your brains.
You have waves all around.
And soon you will find
That oscillations abound.
You’re not on you own if you know what I know.
But YOU are the one who’ll learn how these waves go.

Waves go up and down sine curves, so graph them with care
Though sometimes equations make life less hard to bear.
F equals ma helps you find Diff. EQs
And select t of zero so phi you will lose.

Simple harmonics are the
Pulse of all things.
We study them using
Complex numbers, and springs!

But springiness changes
If life rearranges…

With damping and driving
The amplitude GROWS
If the frequency to
Omega-s goes.

And when resonance happens,
Don’t worry. Don’t stew.
Just know that the max
Is related to Q.

OH! THE PLACES WAVES GO!

Oscillations in springs
And on strings and of light!
With such simple motions,
The waves can take flight!

They don’t lag behind if you add a phase shift:
Delta, we call it, is pi-over-two
Whenever the drive frequency is the square root
Of k-over-m, (omega-s, to you).
But if it isn’t
Then delta is different.

I’m sorry to say so
That nasty
Equations
Can happen to you.

And amplitude, too,
Of the damped-driven kind,
Is less messy and stress-y
With these things in mind.

Tan delta is gamma
Times drive over both
Frequencies squared-minused.
And you’ll then not be loathe

To find amplitude which
Is not so much fun
And deriving this one
is not easily done.

Soon you’ll come to a place where the springs are combined
With pendulum bobs — it will boggle your mind
And the beats mesmerizing will make you cross-eyed.
How can you solve this? Can you even provide
A solution to this question so wide?

Can you split these behaviors into left in right?
Or breathing and pendulum? Or, maybe, not quite?
Can you make any waveform with these normal modes?
The math does work out, and so I suppose
And orthogonal is as orthogonal goes.

You can get so confused
That you’ll start in to race
Through reams of scratch paper at break-pencil pace
And grind on for miles across weirdish wild space,
Headed, I see, toward a most useful place.
The Hilbert Space…

…for waves superposing.
The simple components
Of a pendulum, or a mass-on-spring
Of a water wave, or a loaded string
A cat’s meow sound, or the phone’s shrill ring.
Combining to make waves that go
Wherever it is waves want to go.

Some oscillations in the breeze
The oscillations of the seas
Even the buzzing of the bees.
They all have their own Hilbert Space
Of normal modes which lend some grace –
At least when we’re trying to work out the math
So the gods of normality don’t send their wrath
For wasting so much paper.

Yes! That’s just the thing!

Then these normal modes
help us with waves on a string.
Beaded? Continuous? A solution I bring!

With wave numbers k
We can find all those modes.
Now we’re ready for anything under the sky.
And so we’ll see that waves travel and fly!

Oh, the places waves go! Traveling left! Traveling right!
We can find all the frequencies, even for light.
Because magical things E.M. waves sure are
The travel so easily here, there, afar.
Plane waves! Self-sustaining, move forward at c,
And this is the same ratio as E over B!

Everywhere waves will go
And you know they’ll go far
And you’ve learned all about them,
Whatever they are.

You’ll get mixed up, of course,
You’ll get mixed up
With many strange waves as you go.
So be sure when you guess
A sine-omega-t
To remember ol’ Euler,
Who makes things quite easy.
Just never forget to be dexterous and deft.

And will you succeed?
Yes! You will, indeed!
(98 and three-fourths percent guaranteed.)
Kid, you’ll move SINUSOIDS!

So…
Be your name Buxbaum or Bixby or Bray
Or Mordecai Ali Van Allen O’Shea,
You’re off to more Physics!
And Quantum is waiting.
So… get on that wave!

## 09.18.09

### Ode to Z

Filed under: Harmonic analysis — Travis @

– Jeff Suzuki

Inspired by “Trees,” by Joyce Kilmer.

I think that I shall never see
An aleph set as nice as Z.
A principal ideal domain,
The ring of things that keeps me sane.
Maybe N has better grace
But semigroups should know their place.
To sum a charge or lepton number
Z can count the sheep to slumber.
Units are in Z so rare
Z’s not a field, but what the care?
For Z’s Euclidean, 1, 2, 3,
Unlike Q and R and C.
An Abel group, from 1 came you
Before the alephs, one and two
One to one with N it goes
Countable, as Cantor knows
For R was made by fools like me
But only God can make a Z

The last stanza is a reference to a famous quote by Kronecker: “God made the natural numbers. All the rest are the works of man.”

## 09.17.09

### Lines inspired by a lecture on extra-terrestrial life

Filed under: Harmonic analysis, Science humor — Travis @

– J. D. G. M.

Some time ago my late Papa
Acquired a spiral nebula.
He bought it with a guarantee
Of content and stability.
What was his undisguised chagrin
To find his purchase on the spin,
Receding from his call or beck
At several million miles per sec.,
And not, according to his friends,
A likely source of dividends.
Justly incensed at such a tort
He hauled his vendor into court,
Taking his stand on Section 3
Of Bailey “Sale of Nebulae.”
Contra was cited Volume 4
Of Eggleston’s “Galactic Law”
That most instructive little tome
That lies uncut in every home.
“Cease,” said the sage, “Your quarrel base,
Lift up your eyes to outer space.
See where the nebulae like buns,
Encurranted with infant suns,
Shimmer in incandescent spray
Millions of miles and years away.
Think that, provided you will wait,
Sure to provide you wealth and bliss
Beyond the dreams of avarice.
Watch as the rolling aeons pass
New worlds emerging from the gas:
Watch as brightness slowly clots
To eligible building lots.
What matters a depleted purse
To owners of a Universe?”
My father lost the case and died:
I watch my nebula with pride
But yearly with decreasing hope

From The Observatory 65, 88 (1943).

## 09.15.09

### Mathematical limericks, vol. 7

Filed under: Harmonic analysis, Science humor — Travis @

#### Nonexamples!

pi goes on and on and on …
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed?

Chebychev said it and I’ll say it again:
There’s always a prime between n and 2n.

Man has pondered
Since time immemorial
Why 1 is the value
Of zero-factorial.

Three jolly sailors from Blaydon-on-Tyne
They went to sea in a bottle by Klein.
Since the sea was entirely inside the hull
The scenery seen was exceedingly dull. [FW]

Little Willie was a Chem-E,
Little Willie is no more.
What Willie thought was H2O
Was really H2SO4. [CR]

When calculating polynomial degree,
The minimum value it can be
Is “1″ plus the number of bends.
And remember this too my friends:
The polynomial’s degree is only even
If its graph enters the same side its leavin’.

#### References

[CR] by Crifton Robinson.
[FW] by Frederick Windsor, from The Space Child’s Mother Goose, 1958.

## 09.14.09

### Mathematical limericks, vol. 6

Filed under: Dirty, Harmonic analysis — Travis @

#### Dirty limericks! Let the reader beware!

A mathematician called Able,
Made love to a young girl called Mabel,
Of an old mathematical table.

A mathematician called Babbit
Put some quite simple sums to a rabbit.
The rabbit replied
“I must learn to divide,
With me multiplication’s a habit.”

A mathematician called Cross,
Fell in love with the wife of his boss.
The boss’s reaction,
Suggested subtraction,
He said, “Take her away, she’s no loss.”

A mathematician called Day,
Who was anxious to have it away,
Said the value of X
Turned his thinking to sex,
X times Y was the price he would pay.

A mathematician called Dewar
Whose maths were incredibly pure,
Clamped his penile device
In an engineer’s vice,
Then in microns he measured his skewer.

A mathematician called called Dick
Tried to measure the size of his prick.
But he was enraged
When he found that he gauged
It, not quite the short side of a brick.

A mathematician called Hall,
And the cube of its weight,
Times his pecker, plus eight,
Was four fifths of five eighths of sod all.

A mathematician called Hill,
Had a wife who was not on the Pill.
Though he missed no occasion,
To try multiplication,
The product produced was just nil.

A mathematician called Hyde,
Took a busload of girls for a ride.
And in preparation,
For multiplication,
Each girl forced her legs to divide.

A mathematician named Joe,
Said “Really it just can’t be so;
“My wife, for her sins,
Is going to have twins,
And 2 into 1 doesn’t go!”

A mathematician called Plumb,
Was engrossed in a difficult sum,
And even in bed,
Till his wife said, “For God’s sake, Plumb, come.”

A mathematician called Power,
Calculated his lust in the shower,
But he was nonplussed
When the force of his thrust,
Stopped the water for over an hour.

A mathematician called Rubik,
Has a very strange area pubic.
His balls are both conical,
They look very comical,
With a penis described best as cubic.

A mathematician called Strong,
Got all his conclusions quite wrong.
His value for pi
Was put much too high,
As the average length of his dong.

A mathematician called Week,
Has geometry which is unique.
If A equals B
And B equals C,
ABC is his lower left cheek.

The mathematician Von Blecks
Derived the equation for sex.
He found a good fuck
Isn’t patience or luck
But a function of Y over X.

There once was a log named Lynn
Whose life was devoted to sin.
She came from a tree
Whose base was shaped like an e.
She’s the most natural log I’ve seen.

There once was a man from Rancine
Who invented a fucking machine.
Both concave and convex,
It could serve either sex,
But oh what a bastard to clean!1

There once was a mathematician
Who preferred an exotic position
‘Twas the joy of his life
To achieve with his wife
Topologically complex coition.

The was a young lady called Hatch
So she practiced coition
With a mathematician,
Whose square root was just made to match.

1 was related to me by Greg B.

## 09.13.09

### Mathematical limericks, vol. 5

Filed under: CS silliness, Harmonic analysis — Travis @

#### Computer science contributions

There once was a user named Fred,
Who one day used grep, awk, and sed.
He parsed a huge text stream,
Used regexps to the extreme,
Now his file’s tail is its head.

function createLimmerick(){
var scanning=terriblySlick;
if(lines==5)
&&rhyme=="live"
do(laugh(); performNewTrick();)}

## 09.12.09

### Mathematical limericks, vol. 4

Filed under: Harmonic analysis — Travis @

#### Equations than lend themselves to limericks

Euler’s Equation:

I used to think math was no fun,
‘Cause I couldn’t see how it was done.
Now Euler’s my hero,
For I now see why 0
Equals e pi i + 1.

e raised to the pi times i,
And plus 1 leaves you nought but a sigh.
This fact amazed Euler
That genius toiler,
And still gives us pause, bye the bye.

The Pythagorean Theorem:

A triangle’s sides a, b, c,
With a vertex of 90 degrees,
If that vertext be
‘Tween sides a and b,
The root a-squared plus b-squared is c. [AA]

There are a number of lesser known equations that lend themselves to limerick form:

Equation 1:

A Dozen, a Gross and a Score,
Plus three times the square root of four,
Divided by seven,
Plus five times eleven,
Equals nine squared and not a bit more. [JS]

Equation 2:

Integral v-squared dv
From 1 to the cube root of 3
Times the cosine
Of three pi over 9
Equals log of the cube root of e.

Equation 3:

One over point one-oh-two-three,
When raised to the second degree,
Divided by seven
Then minus eleven
Is approximately equal to e. [AFC]

Equation 4:

Th’integral from e-squared to e
Of 1 over v dot dv,
When raised to the prime
Between five and nine,
Is e to the i pi by 3. [MMB1]

Equation 5:

The integral from naught to pi
Of sine-squared of 2 phi d-phi,
When doubled and then
Not altered again,
Is log (minus 1) over i. [MMB1]

Equation 6:

To find Euler’s Gamma of three,
Integrate to infinity
From zero, dx
x-squared on exp(x),
Or three bang divided by three. [MMB2]

Equation 7:

‘Cause phi-squared less phi, minus 1,
Is exactly equal to none,
The golden mean phi,
Is half of root 5 add on one. [MMB2]

Equation 8:

The square root of minus 2 pi
On th’square root of inverse sine phi;
All that need be done
Is let phi equal one:
It’s twice exp of i pi on i. [AA]

In addition, there are a few figures that lend themselves to limericks:

Figure 1:

If a circle through B, like so,
Has arc AD with center O,
The angle at B,
Wherever B be,
Is half of the angle at O. [MMB1]

Figure 2:

A body with mass m kg
Feels a force of magnitude T.
When its weight t’wards the ground
To speed up at T on m, less g. [AA]

#### References

[AFC] by A. F. Cooper.
[JS] by John Saxon, textbook writer.
[MMB1] by M. M. Bishop.
[MMB2] adapted from M. M. Bishop.

## 09.11.09

### Mathematical limericks, vol. 3

Filed under: Harmonic analysis — Travis @

If you’re mathematically inclined, you’ve probably seen several proofs that 2=1. On of the more popular ones is the following:

This can be expressed as a three-stanza limerick:

If a=b (so I say)
And we multiply both sides by a
Then we’ll see that a-squared
When with ab compared
Are the same. Remove b-squared. Okay?

Both sides we will factorize. See?
Now each side contains a minus b.
We’ll divide through by a
Minus b and olé
a+b=b. Oh whoopee!

But since I said a=b,
b+b=b you’ll agree?
So if b = 1
Then this sum I have done
Proves that 2 = 1.

## 09.10.09

### Mathematical limericks, vol. 2

Filed under: Harmonic analysis — Travis @

#### Number crunching

‘Tis a favorite project of mine
A new value of pi to assign.
I would fix it at 3
For it’s simpler, you see,
Than 3.14159…

If inside a circle a line
Hits the center and goes spine to spine
And the line’s length is d
the circumference will be
d times 3.14159…

If (1+ x) (real close to 1)
Is raised to the power of 1
Over x, you will find
Here’s the value defined:
2.718281…

In arctic and tropical climes,
Taken (mod p) will yield
A full finite field,
As p ranges over the primes.

If n in a Taylor series
Goes 2 to 11 by threes
For n = 1
Convergence is done
‘Twixt 0 and 2, I believe.

## 09.9.09

### Mathematical limericks, vol. 1

Filed under: Harmonic analysis — Travis @

#### There once was a mathematician named x…

Mathematics: of sciences, queen
Has more rules than I’ve ever seen.
There are no exceptions,
Just number deceptions.
On calculators, I am quite keen.

A mathematician confided
That the Moebius band is one-sided
And you’ll get quite a laugh
If you cut one in half
‘Cause it stays in one piece when divided.

A mathematician named Klein
Thought the Moebius band was divine
Said he: “If you glue
The edges of two
You’ll get a weird bottle like mine.”

A go-go lap dancer, a pip,
Was able to peel in a zip.
And died of constriction
Attempting a Moebius strip.

The Moebius strip is a pain,
When you cut it again and again,
But if you should wedge
A large disk round the edge
Then you just get a projective plane.

If you have a cross-cap on your sphere,
And you give it a circle-shaped tear,
And untangle it out
And a Moebius strip will appear!

A mathematician named Crottle
Poured water into a Klein bottle.
That some will run out?”
He replied, “No, I don’t. Quite a lot’ll.”

There was young maiden named List
Whose mouth had a funny half-twist.
She’d turned both her lips
Into Moebius strips…
‘Til she’s kissed you, you haven’t been kissed!

There was a young fellow named Fisk,
A swordsman, exceedingly brisk.
So fast was his action,
The Lorentz contraction
Reduced his rapier to a disc.

A conjecture both deep and profound
Is whether the circle is round;
In a paper by Erdos,
written in Kurdish,
A counterexample is found.

A challenge for many long ages
Had baffled the savants and sages.
Yet at last came the light:
Seems old Fermat was right–
To the margin add 200 pages.

A calc student upset as could be
That his antiderivative didn’t agree
With the one in the book
E’en aft one more look.
Oh! Seems he forgot to write the “+ C”.

Computed the cube of infinity;
But it gave him the fidgets
To write down all those digits,
So he dropped math and took up divinity.

A mathematician called Bird,
Had students who thought him absurd.
There were cries of derision
When he said long division,
Meant one into one made a third.

A mathematician called Rumbold,
One day, quite by accident, stumbled
On the Meaning of Life,
Then went on, for his wife,
To find out why all her apple pies crumbled.

To a tightrope walker named Zekund
The a due to gravity beckoned.
His performance was great
Meters per second per second.

Consider the pitiful plight
Of a runner who wasn’t too bright.
For he sprinted so fast,
That he vanished at last
By red-shifting himself out of sight.

In the near-light speed space-ship I’m in,
I went rocketting off from my twin;
But since I’ve been away
I’ve aged hardly a day
And just look at the state that he’s in!

a vector-valued dominatrix
who divides a phase plane
into pleasure and pain
when she gets hold of more than one matrix. [CG]

Jack the mathematician
The problems he wrote,
He often would gloate,
Sen many a student a-wishin.’ [DG]

There once was a prof, Dr. K.,
Who taught calculus everyday.
From dawn until noon,
Integrating to the moon.
To him, derivatives were okay.

There was a prof named Kowalski
Who taught all this calculus to me.
On the final — no pass;
I must retake the class.
Why, we should all be so lucky. [ES]

Despite all the might fine teachin,’
I can’t help but find myself thinkin’
That Calculus I
Will be much more fun
The second time o’takin’. [MB]

Along cam Sir Isaac Newton
Doin’ mathematical computin’.
One day he contrived
To anti-derive
When findin’ signed areas is suitin’.

#### References

[CG] by Courtney Gibbons (of Brown Sharpie fame!).
[DG] by Dillon Glover.
[ES] by Eric Seifert.
[MB] by Matt Begeman.

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