From: Rohit Khare (rohit@uci.edu)
Date: Sun Mar 19 2000 - 21:07:20 PST
[Since more than one FoRKer has confessed (horrors!) unfamiliarity
with the fable of poor Polly, I post it here once and for all -- RK]
========================================================
Wherein it is related how that Polygon of Womanly Virtue, young Polly
Nomial (our heroine) is accosted by that Notorious Villian Curly PI,
and factored (oh horrors!).
------------------------------------------------------------------------
Once upon a time (1/T) Pretty Polly Nomial was strolling across a
field of vectors when she came to the boundary of a singularly large
matrix. Now Polly was convergent and her mother had made it an
absolute condition that she never enter such an array without her
brackets on. Polly, however, who had changed her variables that
morning and was feeling particularly badly behaved, ignored this
condition on the basis that it was insufficient, and made her way
amongst the complex elements. Rows and columns closed in from all
sides. Tangents approached her surface. She became tensor and tensor.
Quite suddenly, two branches of a hyperbola touched her at a single
point. She oscillated violently, lost all sense of directrix, and
went completely divergent. As she reached a turning point, she
tripped over a square root that was protruding from the erf and
plunged headlong down a steep gradient. When she rounded off once
more, she found herself inverted, apparently alone, in a
non-euclidean space.
She was being watched, however. That smooth operator, Curly Pi, was
lurking innerproduct. As his eyes devoured her curvilinear
coordinates, a singular expression crossed his face. He wondered, was
she still convergent? He decided to integrate improperly at once.
Hearing a common fraction behind her, Polly rotated and saw Curly Pi
approaching with his power series extrapolated. She could see at once
by his degenerate conic and dissipative terms that he was bent on no
good.
"Arcsinh!" she gasped.
"Ho, Ho," he said. "What a symmetric little asymptote you have. I can
see your angles have lots of secs."
"Oh, Sir," she protested, "keep away from me. I haven't got my brackets on."
"Calm yourself, my dear," said our suave operator. "Your fears are
purely imaginary."
"i, i," she thought. "Perhaps he's not normal, but homologous."
"What order are you?" the brute demanded.
"Seventeen," replied Polly.
Curly leered, "I suppose you've never been operated on."
"Of course not," Polly replied quite properly, "I'm absolutely convergent!"
"Come, come," said Curly. "Let's off to a decimal place I know and
I'll take you to the limit."
"Never!" gasped Polly.
"Abscissa!!!" he swore, using the vilest oath he knew. His patience
was gone. Coshing her over the coefficient with a natural log until
she was powerless, Curly removed her discontinuities. He stared at
her significant places, and began smoothing out her points of
inflection. Poor Polly. The algoritmic method was now her only hope.
She felt his hand tending toward her asymptotic limit. Her
convergence would soon be gone forever.
There was no mercy, for Curly was a heavyside operator. Curly's
radius squared itself; Polly's locii quivered. He integrated her by
parts. He integrated her by fractions. After he cofactored, he
performed Runge-Kutta on her. The complex beast even went all the way
around and did a contour integration. Curly went on operating until
he had satisfied her hypothesis. Then, he exponentiated and became
completely orthogonal.
When Polly got home that night, her mother noticed that she was no
longer piecewise continuous, but had been truncated in several
places. But, it was too late to differentiate now. As the months went
by, Polly's denominator increased monotonically. Finally, she went to
L'Hopital and generated a small but pathological function which left
surds all over the place and drove Polly to deviation.
------------------------------------------------------------------------
The moral of the sad story is this:
"If you want to keep your expressions convergent, never allow them a
single degree of freedom."
This archive was generated by hypermail 2b29 : Sun Mar 19 2000 - 21:08:02 PST