Uniformly at Random

Archive for April 2009

Expansions in irrational bases

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Let \beta be an irrational real number between 1 and 2. Let us write the expansion of 1 in base-\beta:

1 = \sum_{n=1}^\infty a_n \beta^{-n},

where a_n \in \{0,1\}.  (We leave it to the reader to speculate as to why one would do such a perverse thing.)  \beta-expansions are not generally unique.  For example, let \varphi = (1 + \sqrt{5})/2 be the golden ratio.  Then 1 = \varphi^{-1} + \varphi^{-2} and 1 = \sum_{n \geq 2} \varphi^{-n}, so .11 and .01111\cdots are both \varphi-expansions of 1. However, there are certain curious irrational numbers \beta for which 1 has a unique \beta-expansion.  Moreover, there is a even a least \beta between 1 and 2 with this property.  This \beta is the unique solution (approx 1.78723…) to

1 = \sum_{n = 1}^{\infty} t_n \beta^{-n},

where t_n is the number of 1’s mod 2 in the binary expansion of n.  (It seems rather bizarre that the parity of 1’s in the binary expansion of n should have anything to do with this problem!)  This \beta is called the Komornik-Loreti constant.  For the proof of this result see the paper of Komornik and Loreti in the American Mathematical Monthly 105 (1998) 636-639.

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Written by uncudh

April 20, 2009 at 3:57 pm

Posted in math

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How a shepherd boy caused the defeat of Napoleon

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Hugo makes the following remark in Les Miserables (Vol. II “Cosette”, Ch. XVIII):

And these things took place, and kings regained their thrones, and the master of Europe was put in a cage, and the old order became the new order, and the light and the shadow of the earth changed places, because on the afternoon of a summer day a peasant boy said to a Prussian in a wood, “Go this way and not that!”

Et ces choses se sont faites, et ces rois ont repris leurs trônes, et le maître de l’Europe a été mis dans une cage, et l’ancien régime est devenu le nouveau, et toute l’ombre et toute la lumière de la terre ont changé de place, parce que, dans l’après-midi d’un jour d’été, un pâtre a dit à un Prussien dans un bois: passez par ici et non par là!

The reference, of course, is to the Battle of Waterloo; the “Prussian” referred to is the commander of the vanguard of Blucher’s army.  Blucher’s timely arrival on the battlefield turned the tide of the battle in favour of Wellington and the Allies, and led to the decisive defeat of Napoleon.

Written by uncudh

April 15, 2009 at 4:21 pm

The Mower to the Glo-Worms

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Andrew Marvell‘s poem “The Mower to the Glo-Worms”:

Ye living Lamps, by whose dear light
The Nightingale does sit so late,
And studying all the Summer-night,
Her matchless Songs does meditate;

Ye Country Comets, that portend
No War, nor Prince’s funeral,
Shining unto no higher end
Then to presage the Grass’s fall;

Ye Glo-worms, whose officious Flame
To wand’ring Mowers shows the way,
That in the Night have lost their aim,
And after foolish Fires do stray;

Your courteous Lights in vain you waste,
Since Juliana here is come,
For She my Mind hath so displac’d
That I shall never find my home.

Written by uncudh

April 11, 2009 at 5:05 pm

Posted in literature

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The Tale of Kullervo

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The “Tale of Kullervo” from the Finnish Kalevala begins with a blood feud between Untamo and Kalervo.  Kalervo’s people are wiped out, save for one pregnant woman whom Untamo takes to be his serf.  When her child, Kullervo, is born, Untamo tries to kill him to prevent Kullervo from taking revenge on him when he grows up, but his attempts to slay the child fail.  Untamo gives up and decides to raise Kullervo as a serf.  However, Kullervo botches every task assigned to him, so Untamo sells him to the smith Ilmarinen.  After Kullervo breaks his knife on a stone baked into a piece of bread by Ilmarinen’s wife, he kills Ilmarinen’s wife and then leaves to seek revenge on Untamo for the death of his kin.  Along the way he discovers that his kin had not all been killed after all.  He seduces a young girl, only to discover afterwards that she was his sister, whom he had supposed to be dead.  His sister then kills herself, as described below (Bosley’s translation):

She just managed to say this
and to tell it once: at once
she tumbled out of the sledge
then ran into a river
into a rapid’s steep foam
into a smoking whirlpool.
There she brought about her doom
there she met her death
found refuge in Tuonela
mercy among the billows.
Kullervo, Kalervo’s son
dashed out of his sleigh
began to weep greatly, to
lament grievously:
‘Woe, luckless me, for my days
a woe, wretch, for my horrors
that I have used my sister
and spoilt her my mother bore!
Woe my father, my mother
woe, woe my honoured parents!
What did you create for me for
and why carry this mean one?
I would have been better off
had I not been born, not grown
not been brought into the world
not had to come to this earth;
doom did not deal straight
disease did not act aright
when it did not kill me, not
lose me as a two-night-old.’

After this tragic incident, Kullervo goes to war and exterminates Untamo and his people.  However, in despair over the disastrous and unlucky course his life has taken, Kullervo decides to kill himself:

Kullervo, Kalervo’s son
snatched up the sharp sword
looks at it, turns it over
asks it, questions it;
he asked his sword what it liked:
did it have a mind
to eat guilty flesh
to drink blood that was to blame?
The sword followed the man’s drift
it guessed the fellow’s chatter
and answered with this word: ‘Why
should I not eat what I like|
not eat guilty flesh
not drink blood that is to blame?
I’ll even eat guiltless flesh
I’ll even drink blameless blood.’

Kullervo, Kalervo’s son
the blue-stockinged gaffer’s child
pushed the hilt into the field
pressed the butt into the heath
turned the point towards his breast
rammed himself upon the point
and on it he brought about
his doom, met his death.
And that was the young man’s doom
the Kullervo fellow’s death—
the end for the fellow, death
for the ill-fated.

One may compare Kullervo’s death-scene with the death of Turin Turambar from Tolkien’s The Silmarillion (or the death of Kullervo’s sister with that of Turin’s sister Nienor):

There he drew forth his sword, that now alone remained to him of all his possessions, and he said: ‘Hail Gurthang! No lord of loyalty dost thou know, save the hand that wieldeth thee. From no blood wilt thou shrink. Wilt thou therefore take Turin Turambar, wilt thou slay me swiftly?’

And from the blade rang a cold voice in answer: ‘Yea, I will drink thy blood gladly, that so I may forget the blood of Beleg my master, and the blood of Brandir slain unjustly. I will slay thee swiftly.’

Then Turin set the hilts upon the ground, and cast himself upon the point of Gurthang, and the black blade took his life.

Written by uncudh

April 9, 2009 at 8:59 pm

The origins of the French and Indian War

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Fred Anderson, in Crucible of War, summarizes the causes of the French and Indian War as follows:

In fact, events had reached a stage at the beginning of 1755 that made war between Britain and France all but inevitable.  The origins of that war lay in a skein of developments so tangled that neither Newcastle nor any other diplomatist in Europe could fully have unraveled, let alone controlled, them.  The decay of the Iroquois Confederacy’s netrality policy and the rising independence of the Indians of the upper Ohio Valley; the surge of Anglo-American traders and land speculators into the region; the fears of the French for the loss of contact by way of the Ohio between the growth of French influence both in the interior of America and on the European Continent; the personalities of Dinwiddie, Duquesne, Newcastle, Cumberland, and even such obscure figures as Washington, Croghan, and Tanaghrisson: in the interaction of all these lay the beginnings of a conflagration that in fact already smoldered on the eastern fringe of the Ohio Valley.  The realignment of the European alliance system, the posting of British and French troops to America, and the dominance of aggressive British politicians would take such comparatively minor episodes and Jumonville’s death and the Battle of Fort Necessity and make of them something much larger, much more dangerous, then even Newcastle at his most pessimistic could have foreseen.  How the clash of tiny numbers of men in a frontier conflict would grow into a world war, how that war would redraw the map of Europe’s empires, and how it would transform the relationship between England and her American colonies—such a chain of events would have defied the most exuberant imagining.  But in a very real sense, as Braddock’s force sailed for Virginia in the first days of 1755, everything had already come to depend upon what it would accomplish, or fail to accomplish, in the depths of the American wilderness.

Written by uncudh

April 8, 2009 at 10:13 pm