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There is a nontrivial
fourth order algebraic differential equation P(y’, y’’, y’’’, y’’’’)=0,
where P is a polynomial in four variables, with integer coefficients, such
that for every continuous function phi, on the line, and for
every positive continuous function epsilon (t), the equation has a C-infinity
solution y for which |y(t) - phi (t)| < epsilon (t) for all real t.”
So there is ONE equation such that YOU name a continuous function and how
close you would like to be to it, and that ONE equation has a solution
that is always that close to the function you named.” (Lee Rubel, Bull.
Amer. Math. Soc., 1981)
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So this guy goes to
a doctor, who says he has Pfaltzmacher’s disease, probably the most contagious
disease known. “We’ll put you immediately in an isolation unit, and
your diet will consist solely of pancakes and pizza,” the doctor said.
“Pancakes and pizza- will that cure it?” “No,” the doctor answered,
“it’s the only food we can get under the door.”
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Call a polynomial p
an n-nomial if p contains at most n nonzero terms, whatever its degree.
Lee Rubel exhibited a single explicit algebraic differential equation of
degree 2n that is satisfied by all n-nomials p, irrespective of the
degree of p. (Nieuw Archief voor Wiskunde, 1988)
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I met Lee Rubel in
1959. I was a newly hatched PhD, and my first job was here at the University
of Illinois. We seemed to hit it off well from the beginning, and
our relationship consisted of equal parts of discussing mathematics, on
the one hand, and telling raucous jokes, on the other.
He was the quintessential
mathematical analyst. Every branch of analysis, one of the broadest areas
of mathematics, interested him intensely. He devised beautiful examples
of phenomena that we scarcely could have imagined existed, and with equal
power, proved strong and elegant general structure theorems about the algebraic
and analytic properties of the beasts that he studied. His mathematical
taste was outstanding. He had no interest in organizations, curriculum
reform, or any of the millions of activities that scholars can get themselves
into when they don’t feel like advancing the frontiers of their subjects.
Lee Rubel was thinking about pushing back those frontiers all the time,
and succeeding.
But he cannot be understood
without considering his marvellous sense of humor and of the absurd, and
his wonderful timing, and the way in which he told his jokes: deadpan -
right to the punch line, only his bushy eyebrows underscoring the ridiculousness
of the situation that was being described.
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A hunter was roasting
a bird over a fire in the woods when a forest ranger burst out of the trees
and said “That’s a loon you’re cooking, and it’s an endangered species
and you’re under arrest!” The man said, “This is no loon - it’s a
duck.” The ranger said “I know a loon from a duck, and you’re under
arrest.” The man pleaded that he really didn’t know, so the ranger
said he would let him off this time. As the ranger was about to leave,
he said “Of course, being a forest ranger, I’ve never tasted loon.
What does it taste like?” The hunter replied “Oh, about halfway between
trumpeter swan and bald eagle.”
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We were close for
those few years. Together we taught ourselves a lot of mathematics, proved
a lot of theorems, and shared a lot of smiles. Then we stayed in touch
for all the years when I was in Philadelphia and he was here [in Illinois].
When his final illness struck him, we suddenly became very close again,
by e-mail, and a large correspondence of jokes and theorems quickly mushroomed.
All of the jokes you are hearing from me today (and dozens more) came to
me from him, in the last months of his life, since last summer. The result
was that while he was dying he spent a lot of energy cheering me
up.
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A Jewish mother is
walking along the beach with her twelve year old son, when the skies darken
and a huge wave sweeps her son out to sea. She looks up at the heavens
and shakes her fist at the dark cloud—Lord, why do you do this to me?
I’ve kept a good kosher house for decades, I go to schul every Friday,
and I obey all your laws.” The skies get very very black and then a second
huge wave comes along and lo! there beside her is her son.
She looks at her son, then looks at the skies and shouts “He had
a hat!”
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On January 24, 1995,
in an e-mail message to me, Lee told me this: “I said to Nina the other
night, `I don’t see why I can’t be permitted to live forever if I just
sit here and smoke my cigar.’”
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There is a single
entire function $f$ such that the set of all derivatives of $f$ is dense
in the space of all entire functions, in the topology of uniform convergence
on compact subsets of the complex plane. - Lee Rubel, 1981.
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Some cats and some
mice died and went to the Pearly Gates, where Saint Peter greeted them
and asked what he could do to make their infinite stay in heaven as pleasant
as possible. The cats asked for some easy chairs and a TV, and the
mice asked for roller skates. Their wishes were granted. After
a few weeks, Saint Peter went to check up, and spoke to the cats first,
asking how they liked things. They were sitting in their easy chairs
in front of a big TV. “It’s really wonderful here!” they said, “and
we especially appreciate the meals on wheels.”
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We have lost a man
who valued, and who practiced, excellence of thought, depth of investigation,
brilliance of insight, and thoroughness of scholarship. We have lost a
man who brightened the lives of all who knew him with the quality of his
mind, with his ability to plumb the depths of his science, with the warmth
of his heart and of his infectious laugh.
He had his share of
the low cards that are in the deck of life.
He made his share
of the high cards: his family, his work, his writings,...
Lee -- we’ll
miss you.
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Remarks prepared for
the memorial service for Lee Rubel, April 16, 1995