My Claim to Fame

On Wednesday night when I was walking back from orchestra, some journalist stopped me and asked me to give a quote for the newspaper. He asked whether I would rather be in class or at the beach on such a nice day. I told him that I would rather be in class since I don’t like beaches. That was quite a silly thing to say, but I didn’t have a lot of time to come up with something decent. Apparently my quote showed up in the newspaper yesterday, but I haven’t verified this independently since I don’t read the newspaper. Scharlemann told me at tea yesterday that he was happy that I had said that since I might have been referring to his class.

Some fellow believes that he has proven that ZF- is inconsistent. However, it appears that the paper is flawed.

And now for a problem (which has not been stated in a well-defined manner yet by me): Suppose we are given a power series centered at 0 with radius of convergence r>0. Now select a complex number z with |z|>r. In terms of the power series coefficients and z (I suppose), what is the density of the smallest number of coefficients we must change in the power series to make the power series converge at z?

Clearly, changing a finite number of terms will never make any difference, and using the formula for the radius of convergence (which I had forgotten when I started thinking about this problem), we can see that if the coefficients are in a geometric progression, then we must modify all but a finite number of terms. What happens in other cases?


About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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4 Responses to My Claim to Fame

  1. your entries are facsinating.

  2. gompers says:

    omgomgomgomgomgomg~!~!~!~!~!~!~! i actually know what the hell you’re talking about!!!!! that’s incredible. (except for the term “destiny”)

    • Simon says:

      Actually the term I used was density. It means the limit of the number I need to change out of the first n terms divided by n. (For example, can I make it work by changing only the even ones, which have density 1/2, or only the ones which are perfect square, which have density 0?)

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