Tuesday, March 29, 2005
Wednesday, March 23, 2005
Twin primes
For a moment I thought I had proved the twin primes conjecture...
The conjecture is that there are infinitely many prime pairs differing by 2. eg. 5,7 ; 11,13 etc.
I started with Euclid's proof of the infiniteness of primes.
Assume for a contradiction that there are finitely many primes. Without loss of generality, assume that there are exactly n primes p1,p2, ... pn in increasing order.
Then, (p1.p2...pn + 1) must be a prime since it has no prime factors, contradicting our assumption.
In the same way, (p1.p2...pn - 1) must also be a prime if n>1.
Doesn't that give us a pair of primes (p1.p2...pn - 1), (p1.p2...pn + 1) for every n that differ by 2?
No, it does not. Why?
The conjecture is that there are infinitely many prime pairs differing by 2. eg. 5,7 ; 11,13 etc.
I started with Euclid's proof of the infiniteness of primes.
Assume for a contradiction that there are finitely many primes. Without loss of generality, assume that there are exactly n primes p1,p2, ... pn in increasing order.
Then, (p1.p2...pn + 1) must be a prime since it has no prime factors, contradicting our assumption.
In the same way, (p1.p2...pn - 1) must also be a prime if n>1.
Doesn't that give us a pair of primes (p1.p2...pn - 1), (p1.p2...pn + 1) for every n that differ by 2?
No, it does not. Why?
Tuesday, March 15, 2005
TV for sale
Some people... anyway, here's an ad I saw recently:
TV for sale. It used to work great until last week when the channel got stuck at number 3. $25.
Hmmmmmmm. $25 per channel. 100 channels. Do the math.
TV for sale. It used to work great until last week when the channel got stuck at number 3. $25.
Hmmmmmmm. $25 per channel. 100 channels. Do the math.
Monday, March 14, 2005
Photo rules
I screwed up. I screwed up big time. There are (at least) 3 fundamental rules for taking good pictures. To consistently take good pictures, one must check these 3 quickly before each click in the following sequence:
1. correct choice of f (aperture) for a desired effect
2. correct exposure
3. hold camera steady
Other rules may be
4. choose proper film for the photo setting eg. 400-1600 for night pictures with people
5. use grad filters if necessary
and many others.
I screwed up in 2. I took a picture of a small bright sunlit flower in front of a dark shaded background. My experience told me that unless I underexposed considerably, my subject (the flower) would be a blur of bright light. I underexposed 2 full stops. The image came out nice. From then on, until the end of the roll, and well into the next one, I kept taking pictures 2 stops underexposed... some of my best compositions came out horribly dark.
1. correct choice of f (aperture) for a desired effect
2. correct exposure
3. hold camera steady
Other rules may be
4. choose proper film for the photo setting eg. 400-1600 for night pictures with people
5. use grad filters if necessary
and many others.
I screwed up in 2. I took a picture of a small bright sunlit flower in front of a dark shaded background. My experience told me that unless I underexposed considerably, my subject (the flower) would be a blur of bright light. I underexposed 2 full stops. The image came out nice. From then on, until the end of the roll, and well into the next one, I kept taking pictures 2 stops underexposed... some of my best compositions came out horribly dark.
Wednesday, March 09, 2005
taking responsibility
I was watching TV and came across a reality show 'Runway' in which some designers compete through their fashion creations. Today's episode was about teamwork, and one of them got eliminated for being the weakest on that count. But that's not really what happened.
It came down to two persons. One had spent several hours crying because she cracked up under pressure. The other simply said that she was slower at sewing than the others and perhaps she herself should be eliminated. The judges agreed, saying that this being an industry, you are supposed to sell yourself, not eliminate yourself. The crying lady got another life.
The decision seemed ironic to me. Here is a person who is willing to take responsibility, who is humble, who instinctively dislikes to pass blame, and who has a kind heart, and she is eliminated on a test of teamwork! I think that I should love to work with such a person as my partner, rather than work with someone who is emotionally immature, or is prone to throwing tantrums.
Somehow, American TV has time and again given me the impression that qualities such as humility, willingness to share responsibility for failures, and not passing blame, are considered weaknesses in this culture. That is perhaps not true is actual work environments, because without these, teamwork is impossible.
Some Buddhist schools, on the other hand, cater to the opposite extreme. Taking responsibility is held sacred. In some of their teaching stories, I came across scenarios where an individual took responsibility quietly and calmly for mistakes that were perhaps not entirely his. And he was served as a model to readers. Isn't is true that taking responsibility for mistakes that you have not committed is a form of dishonesty? It misleads others and just happens to hurt you also. Perhaps the message of the story was that when it comes to taking responsibility, you must check your tendency to get defensive and make excuses. But the example came out as a little extreme to me.
It came down to two persons. One had spent several hours crying because she cracked up under pressure. The other simply said that she was slower at sewing than the others and perhaps she herself should be eliminated. The judges agreed, saying that this being an industry, you are supposed to sell yourself, not eliminate yourself. The crying lady got another life.
The decision seemed ironic to me. Here is a person who is willing to take responsibility, who is humble, who instinctively dislikes to pass blame, and who has a kind heart, and she is eliminated on a test of teamwork! I think that I should love to work with such a person as my partner, rather than work with someone who is emotionally immature, or is prone to throwing tantrums.
Somehow, American TV has time and again given me the impression that qualities such as humility, willingness to share responsibility for failures, and not passing blame, are considered weaknesses in this culture. That is perhaps not true is actual work environments, because without these, teamwork is impossible.
Some Buddhist schools, on the other hand, cater to the opposite extreme. Taking responsibility is held sacred. In some of their teaching stories, I came across scenarios where an individual took responsibility quietly and calmly for mistakes that were perhaps not entirely his. And he was served as a model to readers. Isn't is true that taking responsibility for mistakes that you have not committed is a form of dishonesty? It misleads others and just happens to hurt you also. Perhaps the message of the story was that when it comes to taking responsibility, you must check your tendency to get defensive and make excuses. But the example came out as a little extreme to me.
Sunday, March 06, 2005
The question
The question is not, ‘‘Is there life after death?’’ The question is, ‘‘Is there life before death?’’
Thursday, March 03, 2005
Bouncing ball
quoted from 'The Teaching Stories of Grand Master Wei Chueh'
***
One of the monks was troubled with his wandering thoughts. Not knowing how to deal with them, he had no peace and asked the Grand Master what to do.
The Grand Master said, "Thoughts come, and thoughts go. Just ignore them. Like a bouncing ball, if you don't hit it, it won't bounce."
***
To add to the message of the story above, I quote from an abstract of Zen meditation "Repelling wandering thoughts is like bouncing a ball-the harder you try, the greater the force it bounces back. The right way of dealing with wandering thoughts is to ignore them."
***
One of the monks was troubled with his wandering thoughts. Not knowing how to deal with them, he had no peace and asked the Grand Master what to do.
The Grand Master said, "Thoughts come, and thoughts go. Just ignore them. Like a bouncing ball, if you don't hit it, it won't bounce."
***
To add to the message of the story above, I quote from an abstract of Zen meditation "Repelling wandering thoughts is like bouncing a ball-the harder you try, the greater the force it bounces back. The right way of dealing with wandering thoughts is to ignore them."
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