# Taking balls away: Oz’ Version

This post is based on a comment by Oz to our question about balls with two colors:

“There is an interesting (and more difficult) variation I once heard but can’t recall where:

You have a box with n red balls and n blue balls. You take out each time a ball at random as before. But, if the ball was red, you put it back in the box and take out a blue ball. If the ball was blue, you put it back in the box and take out a red ball.

You keep as before until left only with balls of the same color. How many balls will be left (as a function of n)?

1) Roughly  εn for some ε>0.

2) Roughly $\sqrt n$?

3) Roughly log n?

4) Roughly a constant?

5) Some other behavior

# Answer to test your intuition (18)

You have a box with n red balls and n blue balls. You take out balls one by one at random until left only with balls of the same color. How many balls will be left (as a function of n)?

1) Roughly  εn for some ε>0.

2) Roughly $\sqrt n$?

3) Roughly log n?

4) Roughly a constant?

Here is the collective intuition regarding this problem

# Itai Ashlagi, Yashodhan Kanoria, and Jacob Leshno: What a Difference an Additional Man makes?

We are considering the stable marriage theorem. Suppose that there are n men and n women. If the preferences are random and men are proposing, what is the likely average women’s rank of their husbands, and what is the likely average men’s rank of their wives?

Boris Pittel proved that on average a man will be matched to the woman in place log n on his list. (Place one is his most preferred woman.) A woman will be matched on average to a man ranked n/log n on her list.

We asked in the post “Test your intuition (19)”  what is the situation if there is one additional man, and men are still proposing. This question is based on a conversation with Jacob Leshno who told me about a remarkable paper Unbalanced random matching markets by Itai Ashlagi, Yash Kanoria, and Jacob Leshno. Continue reading

# Andrei

Andrei Zelevinsky passed away a week ago on April 10, 2013, shortly after turning sixty. Andrei was a great mathematician and a great person. I first met him in a combinatorics conference in Stockholm 1989. This was the first major conference in combinatorics (and perhaps in all of mathematics) with massive participation of mathematicians from the Soviet Union, and it was a meeting point for east and west and for different areas of combinatorics. The conference was organized by Anders Björner who told me that Andrei played an essential role helping to get the Russians to come. One anecdote I remember from the conference was that Isreal Gelfand asked Anders to compare the quality of his English with that of Andrei. “Isreal”, told him Anders politely, “your English is very good, but I must say that Andrei’s English is a touch better.” Gelfand was left speechless for a minute and then asked again: “But then, how is my English compared with Vera’s?” In 1993, Andrei participated in a combinatorics conference that I organized in Jerusalem (see pictures below), and we met on various occasions since then. Andrei wrote a popular blog (mainly) in Russian Avzel’s journal. Beeing referred there once as an “esteemed colleague” (высокочтимым коллегой) and another time as  “Gilushka” demonstrates the width of our relationship.

Let me mention three things from Andrei’s mathematical work.

Andrei is famous for the Bernstein-Zelevinsky theory. Bernstein and Zelevinsky classified the irreducible complex representations of a general linear group over a local field in terms of cuspidal representations. The case of GL(2) was carried out in the famous book by Jacquet-Langlands, and the theory for GL(n) and all reductive groups was a major advance in representation theory.

The second thing I would like to mention is Andrei’s work with Gelfand and Kapranov on genaralized hypergeometric functions. To get some impression on the GKZ theory you may look at the BAMS’ book review of their book written by Fabrizio Catanese. This work is closely related to the study of toric varieties, and it introduced the secondary polytopes. The secondary polytopes is a polytope whose vertices correspond to (certain) triangulations of a polytope P. When P is a polygon then the secondary polytope is the associahedron (also known as the Stasheff polytope).

The third topic is  the amazing cluster algebras.  Andrei Zelevinsky and Sergey Fomin invented cluster algebras which turned out to be an extremely rich mathematical object with deep and important connections to many areas, a few are listed in Andrei’s short introduction (mentioned below): quiver representations, preprojective algebras, Calabi-Yau algebras and categories,  Teichmüller theory, discrete integrable systems, Poisson geometry, and we can add also,  Somos sequences, alternating sign matrices, and, yet again, to associahedra and related classes of polytopes. A good place to start learning about cluster algebras is Andrei’s article from the Notices of the AMS: “What is a cluster algebra.” The cluster algebra portal can also be useful to keep track. And here is a very nice paper with a wide perspective called “integrable combinatorics”  by Phillippe Di Francesco. I should attempt a separate post for cluster algebras.

Andrei was a wonderful person and mathematician and I will miss him.

# The Mystery Piano-Player at the Mittag-Leffler Institute

In a previous post I told you about my Mittag-Leffler 2005 experience, and challenged you, readers, to discover the identity of a mysterious piano player. Coming from Yale, I was jet-lagged, an experience which already worked for me once in 1991 when I visited the
Mittah-Leffler institute and  was exploring subexponential variants of the simplex algorithm. In 2005, I came every day to my office at a very early hour in the morning and started working on quantum fault-tolerance.  And very early in the morning somebody was already at the piano. Who was playing the piano for me in the Mittag-Leffler Institute?

Hearing the piano and realizing who my private pianist was on these early mornings reminded me of a very cheerful song that we used to sing in my wife’s family gatherings. The song was introduced by uncle Shimon, husband of my-mother-in-law’s sister Rachel, and he took the lead role.

Shimon start singing: “And what are we going to eat at this feast?”
and we all joined: “A whale and a wild ox, we are going to eat, a whale and a wild ox!”

Wild ox and whale are not kosher (Update: so I, naively, thought), but comprise of the menu of the righteous people in heaven. This was indeed a very cheerful death-defiant song that went on like this:

“And what are we going to drink at this feast?” and we all replied: “Reserve wine, we will drink, reserve wine we will drink, and a whale and wild ox we will eat at this feast!”

And it continued:

“And who is going to taech us the words of the Torah?
Moses our rabbi! Yes, Moses our rabbi is going to teach us the words of the Torah, reserve wine we will drink, and a whale and wild ox we will eat at this feast.

And who is going to tell us words of wisdom at this feast? King Solomon! Yes King Solomon will tell us words of wisdom; Moses our rabbi is going to teach us the words of the Torah, reserve wine we will drink, and a whale and wild ox we will eat at this feast.

And who is going to dance for us, at this feast?
Deborah the prophet! Yes, Deborah the prophet is going to dance for us! King Solomon will tell us words of wisdom; Moses our rabbi is going to teach us the words of the Torah, reserve wine we will drink, and a whale and wild ox we will eat at this feast!

And who is going to play music for us at this feast? King David is going to play music for us at this feast!  Deborah the prophet is going to dance for us! King Solomon will tell us words of wisdom; Moses our rabbi is going to teach us the words of the Torah, reserve wine we will drink, and a whale and wild ox we will eat at this feast!”

Here it is in Hebrew:

ומה נאכל בארוחה זו? שור הבר לוויתן! שור הבר לוויתן נאכל בארוחה זו

ומה נשתה הארוחה זו? יין המשומר! יין המשומר נשתה , שור הבר לוויתן נאכל, בארוחה זו

ומי יאמר לנו דברי תורה בארוחה זו? משה רבנו! משה רבנו יאמר לנו דברי תורה, יין המשומר נשתה , שור הבר לוויתן נאכל, בארוחה זו

ומי יגיד לנו דברי חכמה בארוחה זו? שלמה המלך! שלמה המלך יגיד לנו דברי חכמה, משה רבנו יאמר לנו דברי תורה, יין המשומר נשתה , שור הבר לוויתן נאכל, בארוחה זו

ומי ינגן לנו בארוחה זו? דוד המלך! דוד המלך ינגן לנו, דבורה הנביאה תרקד לנו, שלמה המלך יגיד לנו דברי חכמה, משה רבנו יאמר לנו דברי תורה, יין המשומר נשתה , שור הבר לוויתן נאכל, בארוחה זו

ומי ישמח איתנו בארוחה זו? כל המסובין! כל המסובין ישמחו איתנו, דוד המלך ינגן לנו,  דבורה הנביאה תרקד לנו, שלמה המלך יגיד לנו דברי חכמה, משה רבנו יאמר לנו דברי תורה, יין המשומר נשתה , שור הבר לוויתן נאכל, בארוחה זו

The person who was playing the piano at six o’clock in the morning at the Mittag-Leffler institute was not King David, but rather

Physics, Computer Science, Mathematics, and Foundations’
views on quantum information

Inauguration conference for the Quantum Information Science Center (QISC),
Hebrew university of Jerusalem

Update: The news of our conference have made it to a big-league blog.

Update (July 2013): QStart was a very nice event- there were many interesting talks, and the speakers made the effort to have lectures accessible to the wide audience while discussing the cutting edge and at times technical matters.Streaming video of the talks is now available.

# Test Your Intuition (19): The Advantage of the Proposers in the Stable Matching Algorithm

### Stable mariage

The Gale-Shapley stable matching theorem and the algorithm.

GALE-SHAPLEY THEOREM Consider a society of n men and n women and suppose that every man [and every woman] have a preference (linear) relation on the women [men] he [she] knows. Then there is a stable marriage, namely a perfect matching between the men and the women so that there are no men and women which are not matched so that both of them prefer the other on their spouces.

Proof: Consider the following algorithm, on day 1 every man goes to the first woman on his list and every woman select the best man among those who come to her and reject the others. On the second day every rejected men go to the second woman on his list and every woman select one man from all man that comes to her (including the man she selected in the previous day if there was such a man) and rejects all others, and so on. This process will terminate after finitely many days and with a stable marriage! To see that the process terminate note that each day at least one man will come to a new women, or go back home after beeing rejected from every women (n+1 possibilities) and none of these possibilitie will ever repeat itself so after at most $n^2+n$ days things will stabilize. When it terminates we have a stable marriage because suppose women W and men M are not married at the end. If M is married to a women he prefers less then W or to no women at all it means that M visited W and she rejected him so she had a better men than M.  Sababa!
It turns out that the above algorithm where the men are proposing and being rejected is optimal for the men! If a man M is matched to a woman W then there is not a single stable marriage where M can be matched to a woman higher on his list. Similarly this algorithm is worst for the women. But by how much?

### Random independent preferences

Question 1:  There are n men and n women. If the preferences are random and men are proposing, what is the likely average women’s rank of their husbands, and what is the likely average men’s rank of their wives.

You can test your intuition, or look at the answer and for a follow up question after the fold.