Archive for July 2008

A $1,000 Typo

I resigned from BAE Systems on January 3, 2008. At the beginning of a year, it often takes time for people to adjust to writing the new number. Probably out of habit, someone somewhere wrote that I resigned in January, 2007. The computer at my medical insurance company Cigna got overexcited. It noticed that Cigna paid all my medical bills for 2007, a period for which, according to that creative but premature resignation date, I was ineligible. In its zeal Cigna retracted the money that they had already paid to my doctors for all of my 2007 visits.

In an instant, I became a delinquent; my doctor bills were suddenly more than a year overdue and my credit score probably plummeted. My taxes became suspect, too. I had claimed that I had medical insurance on my Massachusetts taxes, which now Cigna wouldn’t confirm.

While Cigna was racing to get their money back from my doctors, they conveniently forgot that I and BAE Systems paid a lot of money for their insurance. They didn’t hurry to return this money. At one point I was thinking I would be richer if I got back the money paid to Cigna for my insurance and paid the doctors myself.

I had a conference call among Cigna, my former employer and myself. My employer confirmed that I had Cigna coverage until January, 2008, but this was not enough. Cigna insisted that it needed “computer confirmation,” even though it was their shoddy computer work that caused all this trouble.

After many phone calls and conference calls, finally Cigna admitted that I am right and reinstated my medical insurance coverage for the year 2007.

Can you guess what happened next? I received a new bill from my doctor. Cigna reinstated me, but didn’t pay the money back to my doctors. Now there was another round of phone calls and conference calls.

As of today, the time and nerves I spent to resolve this issue exceeded the money Cigna owes to my doctors and I am still waiting for the money to be transferred.

If they are fighting so hard not to pay their debt of $1,000, I wonder about the financial situation of this company. I would definitely consider it very risky to buy Cigna stocks.

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Floating Zipcars

Recently I published my ideas for a Flexible Zipcar Algorithm that allows customers to rent Zipcars one-way. I had a dream the night after I published it: the Zipcar research lab invited me to give a presentation about my algorithm and they asked me a lot of questions. I would like to answer these questions now, but first let me give you a short description of the algorithm:

Some locations are flexible. The number of cars assigned at a flexible location is not fixed, but rather a range between the minimum and the maximum. A Zipster would be allowed to use a car one-way from one flexible location to another, as long the number of assigned cars doesn’t move outside the range.

Question. Zipsters expect to pick up the exact vehicle they reserved. With vehicles changing locations you will break this system. I am operating on the assumption that Zipsters will continue to pick up the exact vehicle they reserved.

Question. What about people who get attached to the cars at their location? You can divide your pool of cars at a flexible location into two groups: fixed Zipcars and floating Zipcars. Only floating cars will be allowed to move. You should advise Zipsters who get attached to their cars to choose fixed cars as the object of their affection. This solution is less flexible, but it resolves your problem.

Question. You designed your algorithm using the min and max number of cars at a flexible location. Where is the desirable number of cars coming from? We need the desirable number of cars to calculate the financial incentives. For example, you can take the absolute value of the difference between the current number of cars at a flexible location and the desirable number of cars. Then sum it up over all locations. We can call this total the distance between the current state and the optimal state. If a one-way trip increases this distance, that Zipster pays extra and vice versa. You can also add a surcharge for all one-way trips to cover the extra expense for parking spaces.

Question. What if cars get stuck? For example, what if we always have the maximum number of cars at Harvard Square and the minimum number of cars at Coolidge Corner? You can always offer free rides from Harvard Square to Coolidge Corner. If every Zipster knows that you have a discount program which is easy to find, your cars will be in the right place in no time. For example, you can color code overstaffed and understaffed locations on your map; or you can add RSS feeds for people who are interested when cheap rides from Harvard Square to Coolidge Corner are available.

Question. Are you saying that instead of hiring drivers to move our cars around we find a much cheaper Zipster who needs to go in the same direction? Yes. If you go even further and offer a token payment or some Zipcar credits, you can have Zipsters driving your cars to oil changes and car washes for you.

Question. Why do you always talk about a small number of flexible locations? I would limit the number of flexible locations for your initial stage, so that for a small investment you get something new, you can experiment, you can observe and study trends in how cars are migrating, you can tune your financial incentives system and you can design and test your web support for this algorithm.

Question. What if someone reserves a car for a one-way trip one month in advance? What happens with the car during this month? If your car – let’s name it “Comfy” — is reserved from the Harvard Square location for a month from now, you need to lock this car into the Harvard Square location for a month. That means, no one can take Comfy for a one-way trip during this month. This looks like a bad case of inflexibility. To resolve it you can have a time-limit for one-way reservations. For example, you might limit one-way reservations to not more than three days in advance.

Question. Suppose we have four cars at the Harvard Square location and the desirable number is three. Then four people reserve four different cars at this location for round-trips on Labor Day. Are these four cars stuck at Harvard Square for a month until Labor Day? I can offer one solution using the earlier idea of fixed and floating cars. You can allow very advanced reservations for fixed cars, but floating cars can only be reserved three days in advance. Another possible solution is that you do the same without dividing your cars into fixed and floating. For example, you allow people to have very advanced reservations for any car which is currently assigned to the Harvard Square location. As soon as three cars are reserved for more than three days in advance, the fourth car is no longer available for very advanced reservations. The fourth car temporarily starts behaving like a floating car, without that being its permanent designation.

Question. Suppose today is Monday and Jack reserves Comfy one-way from Harvard Square to Coolidge Corner for Thursday. To which location is Comfy assigned for three days between Monday and Thursday? On one hand it should be assigned to Harvard Square as it takes a parking space there and we can’t allow another car to take up that space. Also, Comfy is available for reservations before Thursday at Harvard Square. At the same time, Comfy can’t be counted as an extra car anymore as we know that it is guaranteed to be moved. That means that even though Harvard Square actually houses an extra car for these three days, it shouldn’t be flagged as a location with extra cars that need to be reassigned. On the other hand, Comfy will take a spot at Coolidge Corner soon, so we can’t allow other cars to move into that spot. Even though Coolidge Corner doesn’t yet have the complete set of cars, we can’t allow other cars to take up Comfy’s future spot.

Question. It sounds complicated. Can we make it simpler? You can simplify this design for the first implementation. For example, you can only allow immediate one-way reservations. That eliminates any problem with conflicting reservations or an extra car holding up a space for three days. Also, keep in mind that the user doesn’t need to see all this complicated stuff. They are only interested in knowing whether or not they can reserve a car.

Question. What if someone reserves the car one-way then cancels the reservation? This only becomes a problem if someone else reserves the same car for a later time at the destination. In this case, you might institute a very big fee for cancellations of one-way reservations, so that you can either hire a driver or offer a super deal to other Zipsters. Of course, if you allow only immediate reservations for one-way cars this problem will be minimized.

Question. Suppose Jack reserves Comfy for two hours; his starting point is Harvard Square and his end point is Coolidge Corner. Where is Comfy assigned during these two hours? This is a great question. For round-trip Zipcars, you do not care if a person starts a half-hour late or returns the car one hour early. With a one-way car, you need to know when Comfy’s parking space will be available to other cars. To allow Jack to pick up his car as late as he wants or return it as early as he wants, the simplest solution is to assign Comfy for two hours to both locations. That is, no other car is allowed to move into Comfy’s parking spot at Harvard Square until the end of Jack’s rental period and the spot for Comfy should be ready at Coolidge Corner from the start of Jack’s rental period.

Question. You were talking about the simplest solution. Do you mean you have other solutions? I have many ideas, but I prefer to get some feedback from the car-sharing research community before continuing.

Hey, Zipcar! Let’s do it!

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Genetics Paradox

Suppose N mothers live in a city. Half of them have one child and half of them have two children. That means that an average mother has 1.5 children.

Suppose we pick the sexual orientation of every child by rolling dice. Let’s assume that a child has a 10% probability of being homosexual.

The number of mothers with one child who is homosexual is 0.05N. The number of mothers with two children both of them homosexual is 0.005N. The number of mothers with two children with only the first child homosexual is 0.045N, which is the same as the number of mothers of two children with only the second child homosexual. The total number of mothers who have two children with at least one of them homosexual is 0.095N.

Let’s calculate the average fertility of a mother with at least one homosexual child. It is (1*0.05N + 2*0.095N)/(0.05N + 0.095N) = 0.24/0.145 = 1.66. The resulting number — 1.66 — is much bigger than 1.5, the average number of children for a mother.

This means there is a correlation between homosexuality and the fertility of mothers. This suggests that there is a gay gene which at the same time is responsible for female fertility.

But the model is completely random — there can’t be any correlation.

Where is the mistake?

Obviously, you can substitute homosexuality with having blue eyes or math ability or whatever, but I invented this paradox while I was working on my “Fraternal Birth Order Threatens Research into the Genetics of Homosexuality” post. Besides, there is some research on correlation between homosexuality and fertility.

I look forward to your solution to this puzzle.

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Tetramodules in the Air

Three years after I moved to the U.S. and had my second child here, I got an NSF postdoc position at MIT. I had had a long break from my previous research in integrable systems, so I decided to start something new for me — quantum groups. Soon afterwards I invented tetramodules and published a paper about them.

Some time later a friend of mine showed me some papers where my tetromodules had been described long before me. I felt such pain, that I couldn’t work.

I was ashamed that the quality of my paper was compromised. I was very scared that someone might think that it wasn’t original work. I felt worthless for having wasted so much energy on something that was already known.

Afterwards, I discovered that tetramodules appeared in the literature under many other names: bidimodules, Hopf bimodules, two-sided two-cosided Hopf modules, 4-modules, bicovariant bimodules. That meant that many people were reinventing them at the same time. I felt slightly better knowing that I wasn’t alone, but was in fact part of a crowd. Still, my pain impeded my ability to work and do research.

I have just read an article in the New Yorker entitled “In the Air” about some famous discoveries made simultaneously by several people. It made me realize that I am still not over that tetramodules story from 15 years ago.

Being a mathematician requires one to be emotionally strong. You need high self-esteem. Do I need to overcome my emotions or is there a niche for very emotional mathematicians? Perhaps blogging about math is such a niche.

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Fraternal Birth Order Threatens Research into the Genetics of Homosexuality

According to a famous study: ” [E]ach older brother increases a man’s odds of developing a homosexual sexual orientation by 28%-48%.”

This means that sexual orientation is not uniquely defined by genes in the way that our blood type is defined. Indeed, if homosexuality was similar to blood type, the probability of giving birth to a homosexual would be the same for the same parents, independently of birth order. Furthermore, the fact that identical twin brothers quite often have different sexual orientation also supports my claim that it isn’t like blood type. On the other hand, there is some research that shows that there is a genetic component to sexual orientation. This might mean that there are genes that increase a predisposition in a man to become gay or it could be that there is a gay gene which determines their homosexuality for some part of the gay population — or both.

The research into a genetic component of gayness shows that there might be some genes in the X chromosome that influence male homosexuality. It also shows that the same genes might be responsible for increased fertility in females.

By the way, this fascinating research provides an explanation of why “gay genes” — if they exist — do not die out, as evolutionary laws might lead us to expect.

The theory that the probability that someone will be gay is dependent on the fraternal birth order impacts in several interesting ways on the whole field of research into the genetics of homosexuality.

Let us create a theoretical model where homosexuals would be born randomly with a fixed probability if they are the first sons of a woman, and increasing probability for subsequent sons. In this model there would be two interesting consequences:

  • Mothers of homosexual men would be more fertile on average than mothers of men in general. Remember, this random consequence can be misinterpreted as a genetic correlation between homosexuality and fertility.
  • If a homosexual person has a brother, then the probability that this brother is homosexual might be very different from the probability that a random person is homosexual. Again, this might create the suggestion that there is a genetic component, when it is not there.

My point is that all genetics research on homosexuality should take into account these two consequences and adjust for them. That means that it is not enough to show that brothers are more likely to be homosexual in order to prove that there is a genetic component. It needs to show that the correlation between relatives is much stronger than the correlation resulting from the birth order.

For people who are not mathematicians, I will build simple models that illustrate my points. Let us consider some extreme theoretical examples first.

First model. Suppose mothers only give birth to sons and only to one or two sons. Suppose the first son has a zero probability of being gay (which means that first sons are never gay) and the second son has a probability of one of being gay (which means that second sons are always gay). Then all mothers of gay men will have two sons, while mothers of random boys will have somewhere between one and two sons. Another result will be that all gay boys in this model will never have a gay brother.

Second model. Suppose mothers only give birth to one or three sons. Suppose the first son has a zero probability of being gay and the second and the third sons have a probability of one of being gay. Then all mothers of gay men will have three sons, while mothers of random boys will have somewhere between one and three sons. Another result will be that all gay boys in this model will always have a gay brother.

Third model. Let us take some more realistic numbers. Let us consider only the case of women who have one or two boys. Let a be a probability of a woman to have one boy, and correspondingly, 1-a to have two boys. Suppose N is the total number of women in consideration. Suppose x is the probability of the first boy to be homosexual and y is the probability of the second boy to be homosexual. We assume that these probabilities are independent of each other.

Let us first estimate the total number of boys born. It will be aN + 2(1-a)N. The number of homosexuals that are born is expected to be xN + y(1-a)N. The probability of a random boy to be a homosexual is (x + y(1-a))/(2-a).

Let us see what happens with fertility. For a random boy (including both homosexual and non-homosexual boys), what is the average number of sons his mother has? It would be one son for aN boys who are from one-boy families and 2 sons for 2(1-a)N boys who are from two-brother families. Hence a mother of a random boy has on average 2 – a/(2-a) children.

Let’s see what happens with homosexual boys. We have axN homosexual boys from one-brother families and (1-a)(x+y)N gay boys from two-brother families. Hence the average number of sons for their mothers is (axN + 2(1-a)(x+y)N)/(axN+(1-a)(x+y)N) = (ax + 2(1-a)(x+y))/(ax + (1-a)(x+y)) = 2 – ax/(x+y-ay). If we denote by r the ratio y/x, then the result will be 2 – a/(1 + (1-a)r). If r is more than 1, then mothers of gays are more fertile then average mothers. If we assume that y = 1.5x and a = .5, then the average number of sons for a mother of a random boy is 1.667 and for a homosexual boy, it is 1.714, a 3% increase. The impact will be stronger if we take into consideration three-boy families.

Now let us look at brothers in two-son families. There is a total of (1-a)N such families and they have a total of (x+y)(1-a)N gay boys. We have 2xy(1-a)N gay boys who live in families in which both brothers are gay. Hence, the probability of a gay boy who has a brother to have a gay brother is 2xy/(x+y). We saw earlier that the probability of a random person to be gay is: (x + y(1-a))/(2-a).

Let’s look at these numbers more closely. You can easily see that if x = y these two probabilities are the same — as they should be. If x = 0 and y > 0, then a gay person never has a gay brother, suggesting negative correlations with a genetic component. Suppose y = rx, where r is a constant. Then the first number is 2rx/(1+r) and the second number is: x(1+r(1-a))/(2-a). We see that the ratio of these two numbers doesn’t depend on x and is equal to 2r(2-a)/(1+r)(1+r-ar). Suppose r = 1.5, then the ratio is 2.4(2-a)/(5-3a). If a = .5, this ratio is 1.03. So, in this model, for a gay boy who has a brother, the probability that this brother will be gay is 3% higher than the probability that a random man will be gay.

I am so fascinated with the fact that a property that depends on the birth order can create an illusion of a genetic component. I am not discounting the possibility of a genetic component for male homosexuality, but I urge researchers to recalculate their proofs, adjusting for the impact of the fraternal birth order.

To be fair, the female fecundity correlation with male homosexuality was shown not only in mothers of gay men, but also in maternal aunts. Also, the increase in the probability of being gay for a brother of a gay man is very much higher than 3% in my model. That means these researches might survive my critique. Still, they ought to look at their numbers again.

Finally, this discussion is not really about homosexuality, but about any property that depends on birth order. In this case, such a property might imply a genetic component that doesn’t really exist.

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The Symmetries of Things

Building a musnub cubeFinishedThe Symmetries of ThingsThe Symmetries of Things by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss is out. It is a beautiful edition with great pictures.

The first chapter is very nicely written and might be recommended to high school and undergraduate students. It covers symmetries of finite and infinite 2D objects.

The second chapter adds color to the theory. For beautiful colorful pictures with symmetry, there are two symmetry groups: the group that preserves the picture while ignoring its coloring and the group that preserves the picture while respecting its coloring. The latter group is a subgroup of a former group. This second chapter discusses all possible ways to symmetrically color a symmetric 2D picture. The chapter then continues with a discussion of group theory. This chapter is much more difficult to read than the first chapter, as it uses a lot of notations. The pictures are still beautiful, though.

The third chapter is even more difficult to read and the notations become even heavier. This chapter discusses hyperbolic groups and symmetries of objects in the hyperbolic space. Then the chapter moves into 3D and 4D. I guess that some parts of the second and the third chapters are not meant for light reading; they should be considered more as reference materials.

Here are pictures of a musnub cube (multiplied snub cube), built by John H. Conway. It is an infinite Archimedean polyhedron. The description of it is on page 338 and the diagram is on page 339 of the book. This object was glued together from stars and squares. Each corner of the square is glued to a point of one star and to an inside corner of another star. Mathematically, a star is not a regular polygon. If you look at stars with your mathematical eye, each star in the picture is not just a star, but rather the union of a regular hexagon with 6 regular triangles. That means the list of the face sizes around each vertex of a musnub cube officially is represented as 6.3.4.3.3.

The second picture shows a finished musnub cube. You can’t really finish building an infinite structure. Right? It is finished in the sense that John Conway finished doing what he was planning to do: to construct a part of a musnub cube inside a regular triangular pyramid.

Did I mention that I like the pictures in The Symmetries of Things?

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Flexible Zipcar Algorithm

I am now a proud owner of a Zipcard. My old car died a horrible, screaming death recently and I decided to try to go carless as an experiment, hoping to save some money or to lose weight.

Zipcar is the modern way to rent a car. You reserve a car by the hour or by the day through the web, arrive at the site, swipe your card and drive away.

My biggest problem with being a Zipster is that the closest Zipcar location to my home is about one mile away. On second thought, I am losing weight.

But for other Zipsters, the biggest problem is that you have to return the car at the exact location where you picked it up. Obviously, if you allow renters to return their car to a different location the Zipcar company might run out of paid parking spaces in a particular location or, even worse, the cars might migrate to certain places, leaving other locations without any car.

I would like to propose an algorithm that will add some flexibility to where you can return a car, without overwhelming the system.

Here’s how it would work. Suppose we have three cars currently assigned to the Mt Auburn/Homer Ave location. I suggest we name three as a desirable number, but actually allow from three to four cars to be assigned to this location at any particular time.

Now suppose I want to pick up a car at the Mt Auburn/Homer Ave location and to return it to the Somerville Ave/Beacon location. If the number of currently assigned cars to Mt Auburn/Homer Ave location is three (at the lower limit), the Zipcar reservation webpage tells me, “Sorry, you can’t use this location unless you return your car back here,” and shows me the closest location with extra cars. The same goes if the number of assigned cars at the Somerville Ave/Beacon location is at its upper limit. If my starting point has more cars assigned to it than its lowest limit and my destination point has fewer cars assigned to it than its upper limit, then I am allowed to take a car from my starting location and return it to my destination. Zipcar can throw in some financial incentives. If my choice disrupts the most desirable balance of car assignments, I have to pay a fee. If my choice restores the balance, I get a bonus discount.

It would be so cool if zipcars were flexible. I understand that the average cost to the company of parking each car might go up with my flexible algorithm as Zipcar will need more parking spaces than cars. But Zipcar can start implementing this flexibility with a small number of flexible locations. It would be a great feature.

Hey, Zipcar algorithm designers! Can I get a bonus if you implement my algorithm? I can also design a financial incentives formula for you.

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Funny Pictures

Cape Cod Levitation Society

Violators GrilledI updated my funny pictures page with five new pictures. My new favorites are “Cape Cod Levitation Society” on the left and “Grilled Violators” on the right.

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Conway’s Wizards Generalized

Here I repeat the Conway’s Wizards Puzzle from a previous posting:

Last night I sat behind two wizards on a bus, and overheard the following:

— A: “I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”
— B: “How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?”
— A: “No.”
— B: “Aha! AT LAST I know how old you are!”

Now what was the number of the bus?

It is obvious that the first wizard has more than two children. If he had one child then his/her age would be the number of the bus and it would be the same as the father’s age. While it is unrealistic, in mathematics many strange things can happen. The important part is that if the wizard A had one child he couldn’t have said ‘No’. The same is true for two children: their age distribution is uniquely defined by the sum and the product of their ages.

Here is a generalization of this puzzle:

Last night I sat behind two wizards on a bus, and overheard the following:

— Wizard A: “I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age. Also, the sum of the squares of their ages is the number of dinosaurs in my collection.”
— Wizard B: “How interesting! Perhaps if you told me your age, the number of your children, and the number of dinosaurs, I could work out your children’s individual ages. ”
— Wizard A: “No.”
— Wizard B: “Aha! AT LAST I know how old you are!”

Now what was the number of the bus?

As usual with generalizations, they are drifting far from real life. For this puzzle, you have to open up your mind. In Conway’s original puzzle you do not need to assume that the wizard’s age is in a particular range, but once you solve it, you see that his age makes sense. In this generalized puzzle, you should assume that wizards can live thousands of years, and keep their libido that whole time. Wizards might spend so much of their youth thinking, that they postpone starting their families for a long time. The wizards’ wives are also generalized. They can produce children in great quantities and deliver multiple children at the same time in numbers exceeding the current world record.

Another difference with the original puzzle is that you can’t solve this one without a computer.

You can continue to the next step of generalization and create another puzzle by adding the next symmetric polynomial on the ages of the children, for example, the sum of cubes. In this case, I do not know if the puzzle works: that is, if there is an “AHA” moment there. I invite you, mighty geeks, to try it. Please, send me the answer.

In case you are wondering why the wizard is collecting dinosaurs, I need to point out to you that John H. Conway is a superb puzzle inventor. His puzzle includes a notation suggestion: a for the wizard’s age, b for the bus, c for the number of children. Hence, the dinosaurs.

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Sing Away Your Snoring

Singing for Snorers CD

Everyone agrees that you snore because some muscles at the back of your mouth are too relaxed when you sleep. Back in Russia, I heard about exercises for snorers that strengthen those muscles, but I never heard about them here. That is, until now.

Even medicine in this country is oriented towards making money. Simple exercises do not make much money. Here they offer you many devices and pills and even operations to relieve snoring, but not much in the way of exercise. Recently, though, I heard a very simple idea — sing before going to bed. Singing will help tone your throat muscles.

Some people are making money off of singing, too: check out the Singing for Snorers CD.

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