Tanya Khovanova’s Math Blog

I wish I could write four papers in three weeks. The title just means that I submitted four papers to the arXiv in the last three weeks—somehow, after the stress of doing my taxes ended, four of my papers converged to their final state very fast. Here are the papers with their abstracts:

• On k-visibility graphs (with Matthew Babbitt and Jesse Geneson). We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we improve the upper bound on the maximum thickness of bar k-visibility graphs from 2k(9k−1) to 6k, and prove that the maximum thickness must be at least k+1. We also show that the maximum thickness of semi-bar k-visibility graphs is between the ceiling of 2(k+1)/3 and 2k. Moreover we bound the maximum thickness of rectangle k-visibility graphs. We also bound the maximum number of edges and the chromatic number of arc and circle k-visibility graphs. Furthermore we give a method for finding the number of edges in semi-bar k-visibility graphs based on skyscraper puzzles.
• Skyscraper Numbers (with Joel Brewster Lewis). We introduce numbers depending on three parameters which we call skyscraper numbers. We discuss properties of these numbers and their relationship with Stirling numbers of the first kind, and we also introduce a skyscraper sequence.
• Connected Components of Underlying Graphs of Halving Lines (with Dai Yang). In this paper we discuss the connected components of underlying graphs of halving lines’ configurations. We show how to create a configuration whose underlying graph is the union of two given underlying graphs. We also prove that every connected component of the underlying graph is itself an underlying graph.
• Efficient Calculation of Determinants of Symbolic Matrices with Many Variables (with Ziv Scully). Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant algorithms for numerical matrices are often not the fastest for symbolic matrices with many variables. We compare the performance of two algorithms, fraction-free Gaussian elimination and minor expansion, on symbolic matrices with many variables. We show that, under a simplified theoretical model, minor expansion is faster in most situations. We then propose optimizations for minor expansion and demonstrate their effectiveness with empirical data.

Should I eat this piece of cake or not? I will certainly enjoy it very much. What harm will it do? Will this piece increase my weight? Maybe not. The next piece might, but this particular one looks harmless. Even if my weight increases by half a pound, it could be muscle weight. Yes, it probably would be due to muscle weight: I just went out of my house to throw away my garbage and this has to count as exercise.

Do you see the problem? Eating the cake provides an immediate reward, but the punishment is vague and in the far distant future. That is why I got excited when my son Alexey sent me the link to Beeminder, a company that creates an artificial non-vague and not far-in-the-future punishment for eating that piece of cake.

Here is how it works. You give them your target number — in my case my desired weight, but it could be any measurable goal — and the date by which you want to hit it. They draw a yellow path on a weight chart. You must weigh yourself every day. Whenever your weight is above your path, you have to pay real money to the company. Five dollars!

This is a great idea. Suddenly that piece of cake looks threatening. The only problem with using their system is that I have no clue how to lose weight. The company doesn’t provide tools to lose weight: it just provides a commitment device. So it is difficult to stick with the weight-loss commitment without having a proven weight-loss plan.

The truth is that my son sent me the link, I laughed, and forgot about it. Besides, if I ever want to pay money for failing in my commitments, I would rather choose the beneficiary myself. Then I realized that I can use the yellow-road idea to try to lose weight while figuring out what works for me. I call my new plan the Adaptive Diet.

Starting from my actual weight on Day One, I drew a line that represents my target weight, assuming a daily decrease of 0.1 pounds. A deviation of one pound from my target weight on my daily weigh-in is what I call my Yellow Zone. When I am in the Yellow, I continue doing what I was doing before: trying to build new, healthier habits.

If I am more than one pound below my target weight, then I have entered what I call the Green Zone. When I am in the Green, I can allow myself to indulge my cravings. However, when I am one pound above my target weight, I call that the dreaded Red Zone. This Zone has different shades of red. If I am between 1 and 2 pounds above my target weight, I have to eat only apples after 8:00pm. If I am 2 to 3 pounds above my target weight, only-apples time starts at 6:00pm. And so on. Every extra pound above my target weight moves the cut-off time by two hours. That means that if I am 7 pounds above my target weight, I would have to eat apples all day long.

The system has to work: I do not like apples.

My weight used to be my most guarded secret. In general, I am a very open person: I’ll tell anyone anything about me, unless it involves other people. However, there were two exceptions, both of them numbers, interestingly enough: my age and my weight. The closest I came to revealing my weight was with my sister, because we often discuss our similar health issues. Unfortunately, she knows my age, so the only missing number is my weight. I am so tired of my struggle to lose weight, that I’ve stopped caring about keeping the number secret. I am ready to tell it to the whole world.

Let me start from the beginning. I grew up in a country and at a time when men liked plump women. I was never thin, and didn’t have to worry about my weight like my thin girlfriends did. I’ll never forget my high school boyfriend telling me, “Ninety percent of men like fat women, and the other ten percent like very fat women.” When in college I weighed 70 kilograms (154 pounds) and I felt fine. I had my first child when I was 23. I gained 20 kilos while I was breastfeeding, reaching 90 kilos (200 pounds). My husband Andrey kept telling me that he liked Rubenesque women. I wasn’t even slightly concerned about my weight. When we divorced in 1988, I felt that my world was crushed and I didn’t want to go on living. As a result, I lost about 20 pounds.

By 1990 I recovered from my depression, married my next husband, and moved to the US to live with him. The US made me aware of my weight immediately. It didn’t help that Andrey remarried a woman who was the opposite of Rubenesque. From this point on, I wanted to lose weight. After my second child was born, I gained 20 kilograms while breastfeeding, just as I had done with the first child. The result was that I weighed about 220 pounds, much more than I wanted.

I started to look around at what capitalist society had to offer. The pharmacy had many products. I tried Slim Fast, which started to kill my appetite immediately. However, I began to get depressed. The depression felt foreign. As a new mother, I had been very happy before using Slim Fast, and there had been no changes in my life other than consuming Slim Fast. I stopped using it and the depression disappeared. To make sure, I did an experiment. I started using Slim Fast again and the depression reappeared within three days. I stopped it and my depression disappeared. I was so desperate to lose weight that I repeated the experiment. But the result was the same. I stopped using it, and never used any slimming supplement since then. But within that whole process, I lost some weight.

I stayed slightly over 200 pounds for several years. The third time (after the divorce and the Slim Fast) that I lost a lot of weight was when I had my heart broken about 15 years ago. Since then I’ve been slowly gaining weight.

As you can see from my story, I was never able to lose weight when I wanted to. I lost it three times, but I can’t and don’t want to reproduce those circumstances. I actually do not know how to lose weight. For the past ten years I’ve been making changes in my eating habits that I hope, cumulatively, would help me lose weight. I do not buy soda or pizza. I significantly cut my consumption of sweets and starches. I eat more fruits and vegetables. I eat half of what I used to eat in a restaurant. I am still gaining weight.

he only thing I haven’t tried is to be hungry. I am afraid of being hungry. Also I am scared that if I decide on a plan which might result in my being hungry, I will not be able to stick to it. I don’t want to discover that I don’t have enough will power. I am scared to be a failure. I hope that by writing and publishing this I’ll gain the courage to replace my half-measures with a more drastic plan.

Oh! I forgot to tell you: I weigh 245 pounds.

I dropped my blog for two months. Some of my readers got worried and wrote to ask if I was okay. Thanks for your concern.

I am okay. I was consumed by the MIT Mystery Hunt. My team, Manic Sages, won the hunt a year ago, and as a punishment — oops, I meant as a reward — we got to write the 2013 hunt, instead of competing in it. I myself ended up writing about ten problems for the hunt. This was in addition to test-solving about 150 problems my whole team prepared for the hunt.

I could only think about the hunt. My mind was full of ideas for the hunt so I was afraid to write in my blog about something that I might later want to use for my problems. Or even worse, I was nervous that my blog posts might be unconsciously revealing hunt secrets. Moreover, I didn’t want to advertise the fact that I was working on the hunt, thereby drawing people to my blog to scrutinize my interests as they prepared for the hunt.

So I just disappeared.

I apologize; please forgive me.

“I will win the next International Chopin Piano Competition.”

No matter how good I am at positive affirmations, that won’t work: I do not play piano.

I tried to read books on positive thinking, but they made me mistrust the genre. The idea that you can achieve anything by positive thinking makes no sense. For example:

• I can’t win a competition by thinking that I will win. Indeed, everyone can think positively that they are winners, but only one person actually wins.
• I can’t replace work with positive thinking. I will not improve at playing the piano until I take lessons and practice.
• I can’t create the impossible. I can’t turn my eyes blue, no matter how hard I think.

Positive thinking might actually be harmful. I can invest tons of time into trying to change my natural eye color by using my thoughts, when instead I could just use my money and buy some colored contact lenses. Or, if I think myself rich, I might start spending more money than I have and end up bankrupt.

However, perhaps I should not have totally dismissed the idea of positive thinking. While it does have logical inconsistencies, such as those in my examples above, maybe there are ways in which positive thinking is helpful.

First, we should treat these beliefs not as a guarantee, but probabilistically. For example, if you think that you can win the piano competition, the judges will feel your confidence, and may give you slightly better marks.

Second, positive thinking can work, if we choose our affirmations correctly. I recently discovered that I am deceiving myself into believing that I am hungry when I’m not. I should be able to reverse that. I should be able to persuade myself that I am not hungry when I am.

I decided to start small. I tried to persuade myself that tiramisu doesn’t really taste good. Once that seemed to be working, I got more serious. I bought a couple of CDs with affirmations for weight loss.

Unfortunately, they want me to lie down and relax. I do not have time to lie down. I could listen when I am driving or when I am cleaning my kitchen. Hey, does anyone know some good weight-loss affirmations CDs that do not require relaxation?

Samuel Hansen has an unusual profession: he is a mathematics podcaster. He interviewed me for his Relatively Prime podcast titled 0,1,2,3,…, where we discussed my Number Gossip project. The podcast also includes interviews with Neil Sloane, Michael Shamos, and Alex Bellos.

My previous interview with Samuel is at acmescience.com. There I discuss both math education and gender in math issues.

When I listened to myself, I found it strange that I seemed to have a British accent on top of my Russian accent. Did you notice that too?

I am Russian; I know how to play chess. My father taught me when I was three or four. We played a lot and he would always win. I got frustrated with that and one day, when I was five, I didn’t announce my check. On the next move, I grabbed his king and claimed my victory.

He was so angry that he turned red and almost hit me. This frightened me so much that I lost my drive for chess that very moment.

I still understand its beauty and solve a chess problem about once a decade. Look for a cute chess puzzle in my next post.

I would like to report on my weight loss progress. Last time I added two new habits, walking my toy dog every day, and drinking more water from the enticing cute bottles I bought.

I named my stuffed dog Liza and I walk with her every day. I didn’t expect immediate weight loss due to this new regime, because my first goal was to get out of the house every day, even if only for two seconds. The next step will be to increase walking time to ten minutes.

Drinking a lot of water doesn’t work well. I spend too much time looking for bathrooms and panicking that I will not make it. I like the idea of drinking a lot of water, but I am not sure I can hold to it, if you understand what I mean.

Since taking on this challenge, I’ve gained two habits, but I haven’t lost a pound.

Now I’m upping my game. Below is my analysis of why I eat. When I eat, I believe that I am hungry. But looking at this more objectively I think this is not always the case: sometimes there are other reasons. I am listing these other reasons so I can fight them face-to-face. Here we go:

• I eat to finish what is on my plate. My mom lived through World War II in Moscow, and instilled in me a terrible guilt when I throw away food.
• I eat extra when I do not know when my next meal is. I experienced extreme hunger in my childhood, so I try to prevent ever having that terrible feeling again.
• I can’t resist free food. I do not feel comfortable with my financial situation, so saving money gives me an extra push to eat even when I’m not hungry.
• I procrastinate by eating. When I am facing a chore I don’t really want to do, I delay it by eating.
• I crave sugar. It used to be worse.
• I have a problem with delicious food. I think that deep inside I feel that life was unfair to me and this piece of tiramisu will be a small bright spot in my usually rainy life. Therefore I need to grab it and gobble it down before it disappears.

Hmm. That was painful to write. My psychoanalyst taught me that pain means I am on the right track.

Thank you to everyone who helped me to find a host for my Number Gossip website. Some readers and friends even offered me free hosting on their servers. I decided to pay for hosting because I have many specific requirements and that might be a burden on my friends.

On the basis of my readers’ recommendations, I chose Dreamhost as my new webhosting provider. I apologize for the interruption in the flow of the gossip. I know that many people use Number Gossip for birthday gift ideas. I can tell you that on my previous birthday, you could have congratulated me on becoming prime and evil.

I’m trying to lose weight. Many books explain that dieting doesn’t work, that people need to make permanent changes in their lives. This is what I have been doing for several years: changing my habits towards a healthier lifestyle.

This isn’t easy. I am a bad cook; I hate shopping; and I never have time. Those are strong limitations on developing new habits. But I’ve been a good girl and have made some real changes. Unfortunately, my aging metabolism is changing faster than I can adopt new habits. Despite my new and improved lifestyle, I am still gaining weight.

But I believe in my system. I believe that one day I will be over the tipping point and will start losing weight, and it will be permanent. Meanwhile I would like to share with you the great ideas that will work someday.

• Declare some food non-food:
  • I used to keep a kosher kitchen. It was so easy to shop. I didn’t need to go to every aisle in the store because the kosher dietary laws excluded so many foods. Now I’m not kosher anymore, but I like the idea of restricting bad foods, so I created Tanya’s own kashrut rules:
    • Soda is not drinkable.
    • Only dark chocolate deserves to be eaten.
    • Corn syrup, artificial colors and sweeteners are poison.
• Make healthy foods easily accessible:
  • I have Boston Organics fruits and vegetables delivered to me every other week. Initially they all rotted and I had to throw them out, but I am stubborn. Now I’ve learned how to make a turnip salad and how to enjoy an apple. I will soon switch to a weekly delivery.
  • When I’m in a restaurant, I have a rule that I must order vegetables. I do not have to eat them, I just have to order them. But since I do not like things wasted, I end up eating at least some of them. Now I’ve grown to like eggplants and bell peppers.
• Make unhealthy foods less accessible:
  • I buy precut frozen cakes. When I am craving sugar, I defrost one piece. A while ago I would have finished the whole cake the day I bought it, but now, after one piece, I am usually too lazy to defrost another.
  • I buy fewer sweets now. Actually I buy exactly one desert item, as opposed to the half a shopping cart I used to buy. I used to rationalize that I need deserts to serve potential guests. Then I would eat all the sweets myself. Now I’ve decided that my friends will forgive me if I don’t serve desert.
• Engage my friends:
  • Three of my girlfriends and I signed up for the gym together. Without them, I would have dropped the gym a long time ago. Natasha’s call inviting me to yoga often is the extra push that I need. Now, several years later, the habit is formed and when necessary I go alone.
  • Introduce other good habits:
    • I have a separate computer for games. I put it on top of my bookcase, so I have to stand while playing. This way I can’t play for too long, and burn extra calories at the same time.

  I have many other ideas that for different reasons haven’t yet become habits. So I am thinking about tricks to turn them into habits.

  • Start every meal with water.

  I keep forgetting to start my meals with water. Besides, I do not like plastic bottles. So now I’ve bought glass bottles with protective sleeves to carry in my car and my bag conveniently. They look so cool that I enjoy sipping from them.

  • Exercise every day.

  I never exercise in the mornings, because I want this time for mathematics. But in the evenings I am often too tired and skip my scheduled gym sessions and dance classes. I often spend the whole day inside in my pajamas. So to help me to exercise daily, my friend crocheted a small toy dog for me. Now I pretend that it’s a real dog that needs to be walked every day.

  I have many more ideas, but I gotta run now. I need to walk my dog.