Thursday, November 29, 2007

Ulm Theorem

One more name from the name index mention in the previous posts was "Ulm". The text spoke about "Ulm theorem", "Ulm sequence". Again, with no initial. Internet search gives "Ulm" most often related to "Ulm University", and "theorem" is so common word. Nevertheless, rather quickly I found:

H. Ulm proved in 1933 that the structure of a reduced countable Abelian p-group is completely determined up to isomorphism by a sequence of invariants called the Ulm invariants. The original methods he invented for the computation of these invariants and the construction of isomorphisms require a transfinite iteration whose length, depending on the group, can be any arbitrarily large countable ordinal. One may therefore ask whether there is an alternative algorithm that requires only transfinite recursions with bounded countable lengths.

[Carry Lee, Borel Isomorphism Relations of Countable Reduced Abelian p-Groups, Oct 14, 1997]

Further search led to the review of the original article [Ulm, H. Zur Theorie der abzählbar-unendlichen Abelschen Gruppen. (German) Math. Ann. 107, 774-803 (1933) and others] in Zentralblatt.

The most interesting finding was the book:
Franz Lemmermeyer und Peter Roquette, Helmut Hasse und Emmy Noether, Die Korrespondenz 1925-1935, Universitätsverlag Göttingen, 2006.

In the section "Kurzbiographien" we read:

Ulm, Helmut (21.06.1908 Gelsenkirchen – )
Studierte 1926–1930 in G¨ottingen, Jena und Bonn; dort promovierte er 1930, und wurde 1947 als Professor an die Universit¨at M¨unster berufen.

[about him]
R. G¨obel, Helmut Ulm: His work and its impact on recent mathematics, Contemporary mathematics AMS 87 (1989), 1-10.


Lyrics
Joe Dolan
It's You, It's You, It's You
[Brandmaier]

It's You, It's You, It's You
The only one i want
The woman I've been searching for
And if you look at me
You'll see a lonely man
Who's been too long without a home

So why don't we just get together
And try to live our lives as one

It's You, It's You, It's you
The only one i need
To fill my empty heart

I've never been in love before
They tell me that it's beautiful
So darling let me rock
Forever in your arms

It's You, It's You, It's You
The only one I want
The woman I've been searching for
And if you look at me
You'll see a lonely man
Who's been too long without a home

So why don't we join hands together
And be a happy family

It's You, It's You, It's You
The only one I want
To be the mother of my child

I've never been in love before
They tell me that it's beautiful
So darling let me rock
Forever in your arms

It's You, It's You, It's You
The only one I want
The woman I've been waiting for



Monday, November 26, 2007

Chu Spaces

Another name in the name index I worked on was "Chu". It was related to "Chu spaces". The article "Chu space" in Wikipedia says:

The Chu space concept originated with Michael Barr and the details were developed by his student Po-Hsiang Chu, whose master's thesis formed the appendix of the book M. Barr, *-Autonomous categories, Springer-Verlag Lecture Notes in Mathematics 752, 1979.

The relation between the author of this concept and the concept's name is very interesting.

M. Barr, in his article "The Chu construction: History of an idea", Theory and Applications of Categories, 17:1, 10-16 (2006), writes:

<...>
It seemed clear that this gave *-autonomous category, but there were a number of unpleasant details to be verified. Since my student, Po-Hsaing Chu needed a master's project, so I asked him to verify them, which he did in [Constructing *-autonomous categories, M.Sc. thesis, McGill university, 1978] [APPENDIX to M. Barr, *-Autonomous categories]"

I now had expanded from six to infinity the repertory of known *-autonomous categories. At this point, the construction, and in fact the whole idea of *-autonomous categories was more-or-less forgotten, not least by me.

In the earlier paper M. Barr [The Chu construction, Theory and Applications of Categories, 2:2, 17-35 (1996)] gave the history of Chu construction (Section 6 "The true history of Chu categories").

<...>
So I wrote down the obvious definitions of morphism. It then occurred to me wonder if the resultant category was closed or monoidal and quickly discovered that it was both, modulo the checking of a morass details. Since Chu needed a topic for his master's degree, I set him this task and the appendix to [Barr, 1979] was the result. But the story does not end there. Or rather, it would have ended there since neither I nor anyone else found any interest in this rather formal construction. But the year 1987 saw a renewed interest in *-autonomous categories as models of Girard's linear logic and then Vaughan Pratt and his student Vineet Gupta rediscovered the Chu construction and begun to study it in earnest.
<...>
Probably the main conclusion to draw from this history is that, as with most mathematical discoveries, it is a mistake to attribute it to any one person. It was born out of need, with a number of midwives, but no real discoverer.

The APPENDIX to the monograph [Barr, 1979] (of some thirty pages) seems to be the only publication by Chu on "Chu spaces".
So, the Chu spaces are named after Po-Hsaing Chu.


Lyrics
Joe Dolan
You're Such A Good Looking Woman

When God created a woman for me
he must have been in a beautiful mood
to show the world what a woman could be
when he created a woman like you.
He made the sunshine right out of your eyes
he made the moonglow all over your hair
he put a soft summer breath in your sighs
so you could breath summer into the air.

Up there in heaven I bet the are mad
I bet somebody will want to know why
the most incredible angel they had
was found to be quite unable to fly
D'you know what they had forgotten to do
up there where they make all those heavently things
they made an angel as Lovely as you
but the'd forgotten to fit you with wings.

R E F R A I N :

Oh me oh my you make me sigh
you're such a good looking woman
when people stop and people stare
you know it fills my heart with pride
you watch their eyes they're so surprised
they think you've fallen out of heaven
and if you listen to what they're talking about
who's walking about with an angel at his side.

Joe Dolan (born Joseph Francis Robert Dolan, 16 October, 1943 in Mullingar, County Westmeath, Ireland) is a singer of easy listening songs.

Monday, November 19, 2007

Study Theorem

My task was to compile the name index for a book. The sentence "The study of finite dimensional $U_k$-modules $M$ is dominated by the classical result due to Lie and Study ... " offered two entries to the name index: 'Lie' and 'Study'.

The sentence spoke about the 'classical result of Study' or the 'classical theorem of Study', etc. The initials should be added, too. Simple search for phrases with the word 'Study' ended with million and more of links to various documents which could not disclose the initials of the name 'Study'.

Also, some of links led to 'Study's theorem on curves' while the sentence in the book was related with something close to the Lie groups, algebras, modules, representations, etc.

Looking for 'Study theorem' in Wikipedia the Wikipedia page Wikipedia:Missing science topics/Maths27 directed me to Google book search for 'Study's theorem'. The Google book search gave about 160 links, three of which were useful enough.

In the pages of books shown via the Google book search I read the following.

"Essays in the History of Lie Groups and Algebraic Groups" by Armand Borel

p. 11:
In 1896, G. Fano, who knew about Study's theorem through [LE] and was surely not aware of Cartan's proof, maybe not even of Cartan's thesis, gave an entirely different one in the framework of algebraic geometry, using the properties of "rational normal scrulls" [F].

p. 31:
Another push into Lie groups came, in a way, again, from the tensor calculus in [WI], but for a different reason. In 1923 E. Study, a well-known expert in invariant theory for over thirty years, published a book on invariant theory [St2].


I already could conclude that the theorem was 'E. Study theorem', and the name was 'E. Study', but it was interesting to read more.

"Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1826-1926" by Thomas Hawkins
[Chapter 7 "Lie's School and Linear Representations"]

Three Privatdozenten at Leipzig, Friedrich Engel, Friedrich Schur, and Eduard Study, turned their research interests to aspects of Lie's theory.
<...>
Then by Study's complete reducibility theorem for $\mathbf{sl}(2,\mathbb{C})$ all the possibilities for $\mathfrac{m}$ could be specified. That is, Study's theorem implies
\begin{equation}
\mathbb{C}_{n+1}=\mathcal{V}_0\oplus \mathcal{V}_1\oplus\cdots\oplus \mathcal{V}_k\oplus,
\end{equation}
where $\mathfrac{m}$ leaves each vector subspace $\mathcal{V}_i$ invariant but leaves no nontrivial subspace of $\mathcal{V}_i$ invariant for $i>0$ and acts trivially on $\mathcal{V}_0$.

<...>
According to Kowalewski, Engel had called his attention to Fano's paper [1896], discussed in Section 3, where Study's complete reducibility theorem for $\mathbf{sl}(2,\mathbb{C})$ was already emplyed to solve a different problem.
<...>
Kowalewski remained in Leipzig as an instructor (Privatdozent) until 1901, when he obtained a position as an asistant professor (ausserordentlicher Professor) at the University at Greifswald, where Study was professor.
<...>
In 1904 Study moved to a professorship at Bonn, and Kowalewski followed him there in 1905. (Engel, who had never held a tenured professorship at Leipzig, became Study's replacement at Greifswald)


I see that the theorem in question is called "
Study's complete reducibility theorem". The Google book search for this phrase led to

"Hermann Weyl, 1885-1985: Centenary Lectures" by Chen Ning Yang, Roger Penrose, Armand Borel, Komaravolu Chandrasekharan
Armand Borel: "Hermann Weyl and Lie Groups"

Since E. Study was somewhat of a villain in the 1923 incident related earlier, let me add as a counterpart that he was well aware of this problem around 1890 and had brought the first contribution to it. In fact, S. Lie reports in [L: 785-8] that Study has proven full reducibility (phrased however differently, in terms of projective representations) for $\mathfrac{sl}_2$ in
an unpublished manuscript and that it was quite sure it would be frue more generally for $\mathfrac{sl}_n$. In a letter to S. Lie (December 31, 1890), referred to in [Hw], Study even goes as far as conjecturing it should hold for a simple or semisimple Lie algebras. To both of them, this was an important problem. The manuscript was not published, apparently because the proof appeared too complicated and simplifications were hoped for.


The Ultimate Study Theorem

Let's derive the following equations:
Assume the following equations are true:

Study = Not Fail
Not Study = Fail

Then:
Study + Not Study = Not Fail + Fail
Study (1 + Not) = Fail (1 + Not)

Therefore:
Study = Fail

It was also found as a side effect ...



Lyrics

Good Hearted Woman
Willie Nelson
Waylon Jennings

A long time forgotten
are dreams that just felt by the way.
And the good life he promised
ain't what she's living to day.
But she never complains of
the bad times or bad things he's done
Lord.
She's just talks about the good times
they've had and all the good times to come.

She's a good hearted woman
in love with a good timin' man.
She loves him in spite of his ways
that she don't understand.
Through teardrops and laughter

they'll pass though his world hand in hand.
A good hearted woman
lovin' her good timin' man.

He likes the night life

the brightlights and good timin' friends.
when the party's all over
she welcome him back home again.
Lord knows she don't understand him
but she does the best that she can.
'Cause she's a good hearted woman
she loves her good timin' man.

She's a good hearted woman ...

Wednesday, November 14, 2007

Ruby And All That

I read the post "Is Ruby the New VB?" on TheServerSide Interoperability Blog with interest. Even more the later discussion "Interop Blog: Is Ruby the new Visual Basic?".

And I decided to put some excerpts on my notebook.

Posted by: Paul Beckford on October 31, 2007

The reason why both Sun and Microsoft are jumping on to Ruby is because they both have a lots of clever people that really understand OO and functional languages like Smalltalk and Lisp. Sun did a lot of research on OO, specifically with Self, and Microsoft has a massive research budget and has been doing lots of interesting stuff with functional programming ideas such as LINQ.

I remember reading articles by people who were struggling to build OO component frameworks in the early 90's with C++. Who remembers Taligent, OpenDoc and SOM? CORBA and COM are other examples, and let's not forget 'OO' Operating System attempts like Pink. All these things failed because C++ was not OO and wasn't the right tool for the job. Yet marketing bosses insisted that an inappropriate hybrid language derived from C was used because that was the incumbent technology of the day.

What I find interesting is that a Japanese home hobbyist has knocked up a true OO language in his spare which has gone on to be "adopted" by "Software" companies like Sun and Microsoft with multi-million dollar research budgets :^).

What is the lesson to be learned here? Firstly vendors will sell you what ever it is they think you will buy, and secondly we end up with the technology we deserve.

As far as OO is concerned the mainstream is mostly in the dark ages and IMHO the average Java programmer still doesn't fully appreciate what OO is all about (i.e message passing). Without open source language efforts like Python, Squeak and Ruby mainstream programming was destined to stay in the dark ages for an eternity, and vendors would happily have carried on selling us so called "OO" manure like EJBs 2.1.


Posted by: Paul Beckford on October 31, 2007

<...> Ruby is even more like Smalltalk. In Ruby only message sends are supported and there are no hard wired function calls allowed at all; and of course Ruby as uniformity where everything is an object. This explains why Ruby can support advanced OO abstraction techniques like Mixins.

My main point is that both Sun and Microsoft know all this stuff and they have known it for quite some time (say about 30 years :^)). It is for these reasons that they are now jumping onto the Ruby bandwagon (higher programming abstraction and higher developer productivity).

The only people that seem to be in the dark about this stuff is the mainstream programming community who have seemingly bought the misleading OO marketing spiel pushed out by these very same vendors hook line and sinker.


Posted by: Paul Beckford on November 02, 2007

<...> Sun and Microsoft are aware that a popular Smalltalk-like language could gain significant market/brain share. I think they see this as a potential threat to the current encumbent technology base (Java/J2EE and C#/.Net) and hence to themselves.

As such, they want to ensure that they have a signifcant stake in "what ever comes next". From their perspective this is just sound business sense.

I'm sure that both Microsoft or Sun would love to "own" mainstream software development by having everyone use C# or Java, but with the rise in popularity of open source languages I think they both realise that this now unlikely.

In addition to this as pointed out by Frank and Cedric, these corporations don't speak with a single voice. Some within these organisations have wanted to promote alternative languages for years. IMO it has been the business/marketing sides of these companies that have put out the "golden hammer" message when it comes to their own "proprietary" programming languages, and have chosen to promte an interpretation of OO that places their languages in a favorable light.

I don't blame them for this, after all they are in business to make money for their shareholders. I just wanted to draw attention to how we the developer community have done a poor job of maintaining any semblance of control over the types of tools we are obliged to use on a daily basis.

Posted by: Huw Collingbourne on November 05, 2007

Dolphin Smalltalk! Now you and I are definitely thinking along the same lines. I love(d?) Dolphin Smalltalk and I was very disappointed when Object Arts (who made it) decided to stop future development. It constantly amazes me how many great ideas were put into the 'original' Smalltalk towards the end of the '70s/early '80s and which have still not been taken up by modern languages and IDEs.

One of the things I find frustrating about many Ruby programmers' coding style is that they create 'flat' class hierachies - i.e. everything descends from Object - and populate them with long and complex methods, whereas Smalltalk programmers typically create dense hierachies (with many levels of descent) but very small methods (as only the differences from ancestor methods need to be coded). To a large extent, I think this is explained by the fact that the Smalltalk language is provided with an environment optimized for OOP so it's easy to create complex family trees whereas, until quite recently, there were no decent IDEs for Ruby at all.

But there is still a long way to go before Ruby coding becomes as OOP-friendly as Smalltalk. It does seem bizarre in the 21st Century to be looking back to a language/IDE produced in the '70s and '80s but I really do think that there is much about Smalltalk that still seems surprisingly 'modern'.


Lyrics
Eddy Arnold (1948)
Bouquet of roses

I'm sending you a big bouquet of roses
One for every time you broke my heart
And as the door of love between us closes
Tears will fall like petals when we part.

I begged you to be different
But you'll always be untrue
I'm tired of forgiving
Now there's nothing left to do.

So I'm sending you a big bouquet of roses
One for every time you broke my heart.

--- Instrumental ---

You made our lover's lane a road of sorrow
Till at last we have to say goodbye
You're leaving me to face each new tomorrow
With a broken heart you taught to cry.

I know that I should hate you
After all you've put me through
But how can I be bitter
When I'm still in love with you.

So I'm sending you a big bouquet of roses
one for every time you broke my heart...

Tuesday, November 6, 2007

Self-Expression and Communication

I began to realise that self-expression is a solitary activity, in which a single person attempts to put into words thoughts that are in its own mind. Communication, on the other hand, is a social activity, in which the person attempts to put into words thoughts that it wants to be in another person's mind.

I learned that to communicate effectively, you need to express yourself as clearly and as simply as possible. You need to express yourself clearly, so that your readers (or listeners) understand what you intend and do not understand something else. And you need to express yourself simply, so that your readers do not expend unnecessary effort to extract a useful form of your intended meaning. I eventually convinced myself that both of these characteristics of effective communication have a logical interpretation.

From "From Mathematical Logic, to Natural Language, Artificial Intelligence, and Human Thinking (A Short Essay in Honour of Andrzej Mostowski)" by Robert Kowalski



I find that my previous post has some common points with this quote. At least in that there is indeed a problem of expression (art) vs. communication.



Lyrics

It's Your Song
Garth Brooks

Standing in the spotlight
On such a perfect night
Knowing that your out there listening
I remember one time
When I was so afraid
Didn't think I had the courage
To stand up on this stage
Then you reached into my heart
And you found the melody
And if there ever was somebody
Who made me believe in me
It was you
It was you

It was your song that made me sing
It was your voice that gave me wings
And it was your light that shined
Guiding my heart to find
This place where I belong
It was your song

Every night I pray
Before the music starts to play
That I'll do my best and I won't let you down
And for all the times I've stood here
This feeling feels brand new
And any time I doubt myself I think of you

'Cause It was your song that made me sing
It was your voice that gave me wings
And it was your light that shined
Guiding my heart to find
This place where I belong
It was your song

Dreams can come true
With God's great angels like you

It was your song that made me sing
It was your voice that gave me wings
And it was your light that shined
Guiding my heart to find
This place where I belong
It was your song
It was your song
It's always been your song

Sunday, November 4, 2007

The Century of China

More than once he told me that the next century would be the century of China.

From "A Very Personal Recollection" [on Andrzej Mostowski] by Hidemitsu Sayeki


Lyrics
Bay City Rollers
I Only Wanna Be With You

I don't know what it is that makes me love you so
I only know I never want to let you go
Cause you started something
Oh can't you see
Ever since we met you've had a hold on me
It happens to be true
I only want to be with you

It doesn't matter where you go or what you do
I want to spend each moment of the day with you
Look what has happened with just one kiss
I never knew that I could be in love like this
It's crazy but it's true
I only want to be with you

You stopped and smiled at me
And asked me if I cared to dance
I fell into open arms
And I didn't stand a chance

Now listen honey
I just want to be beside you everywhere
As long as we're together honey I don't care
Cause you started something
Oh can't you see
Ever since we met you've had a hold on me
No matter what you do
I only want to be with you

You stopped and smiled at me
And asked me if I cared to dance
I fell into open arms
And I didn't stand a chance

Now hear it honey
I just want to be beside you everywhere
As long as we're together honey I don't care
Cause you started something
Oh can't you see
Ever since we met you've had a hold on me
No matter what you do
I only want to be with you
Oh no matter, no matter what you do
I only want to be with you

Imagine Invisible Inhearable Invisible

Last night looking at the nice symphony orchestra and listening nice music...

You can close your eyes and still hear music. Nothing strange. Imagine you cannot hear. (Some people do not hear.) You'll see then movements of hands, heads, etc. of orchestra players only. To my mind these movements were even more strange without movements of the conductor.

Not strange enough? Imagine then that you cannot see instruments. There are two possibilities: to hear music or not. Suppose you hear. Perhaps you will find that movements of orchestra players are somehow meaningful under (or with) music.

Suppose you cannot hear music and cannot see instruments. You see only movements of players (and the conducter) without instruments. Will you find a difference between, say, Dvorak's Ninth Symphony and Beethoven's Ninth Symphony, well, except for duration? Can this difference be measured or expressed in any means?

Maybe you enjoy sports more that music. Imagine basketball or baseball (you can insert the name of your favorite play here) teams playing with a ball which is not visible to spectators only.

It should look as a kind of dance, but without music. Can you imagine music suitable for such a dance? Can other teams learn to dance (to play) under such a music, but with no real ball? Would it look alike to very modern ballet?


Lyrics

Last Night
Travelling Wilburies

Verse 1:
She was there at the bar, she heard my guitar
She was long and tall, she was the queen of them all
Last night, thinking about last night
Last night, thinking about last night

Verse 2:
She was dark and discreet, she was light on her feet
We went up to her room and she lowered the boom
Last night, thinking about last night
Last night, thinking about last night

Bridge:
Down below they danced and sang in the street
While up above the walls were steaming with heat
Last night, thinking about last night
Last night, thinking about last night

Verse 3:
I was feeling no pain, feeling good in my brain
I looked in her eyes, they were full of surprise
Last night, talking about last night
Last night, talking about last night

Bridge 2:
I asked her to marry me she smiled
and pulled out a knife
The party's just beginning she said,
it's your money or you life
Last night, talking about last night
Last night, talking about last night

Verse 4:
Now I'm back at the bar, she went a little too far
She done me wrong, all I got is this song
Last night, thinking about last night
Last night, thinking about last night