If I understand him correctly, such a uniform bound on the multiplicity of an eigenvalue lambda isn’t known for any lambda other than 0. (Though a bound of the form o(dim(S_k)) is given for nonzero lambda by Frank Calegari on his blog.)

The bound o(dim(S_k)) is due to Serre which is proved in the following paper and he discusses some arithmetic applications:

There are two natural normalizations of the Hecke operators T_p which are important from the arithmetic and analytic point of view. I denote them by the arithmetic and analytic normalization.

1) Arithmetic normalization: Let f be a Hecke weight k modular form of fixed level N where gcd(N,p)=1. We normalize T_p so that

-2p^{(k-1)/2}<a_f(p)<2p^{(k-1)/2} (this is the Ramanujan bound)

where a_f(p) is the p-th Hecke eigenvalue of f. It follows from the argument of Frank Calegari on his blog, that the number of weight k modular forms f with a_f(p)=\lambda is bounded by

O(v_p (\lambda)^2) (*)

where v_p is any p-adic valuation defined on \bar{Q}. This bound is independent of the weight k! However, by Gouvea conjecture on the distribution of the slopes of the modular forms, we expect $v_p(\lambda)/k$ is equidistributed with respect to the uniform measure on $[0,(1/p+1)].$ So, (*) gives $m_p(\lambda,k,N) \ll k^2$ for most of the Hecke eigenvalues which is unfortunately worse than the trivial bound $m_p(\lambda,k,N) \ll k$. In his recent post, Frank explains this and amazing p-adic version of the Weyl law:

https://galoisrepresentations.wordpress.com/2018/10/16/more-or-less-opaque/

2) Analytic normalization: Define \lambda_f(p):=a_f(p)/p^{(k-1)/2}. Hence

-2<\lambda_f(p)<2

In this normalization, there is not any better bound than Serre's bound O(k/log(k)) on the number of Hecke modular forms f where

\lambda_f(p)=\lambda.

So, as you pointed out: such a uniform bound on the multiplicity of an eigenvalue lambda isn’t known for any lambda other than 0.

]]>“I finished the novel last night. It was an engaging story, despite the problems I had with the uneven writing quality, especially in the middle third. (There are also complaints about historical accuracy, and I noted a very narrow view of the revolution — it almost appears that the Defarge’s are running the whole show.) The heroic ending seems uniquely romantically British, and even Shakespearian, in character.”

[Actually, I later read a Dumas novel with a very similar kind of ending. I’m sure that someone borrowed from the other; I haven’t taken the time to find figure out who.]

I’m also interested because after reading Les Miserables unabridged, and several of Dumas’ multi-volume works (totaling thousands of pages each), I’m looking for big novels that I can get lost in for a while. I still have Proust on my list, but I’m hung up on which edition to read.

The original edition of Nicholas Nickleby featured a Seguin as an opera singer. Seguin was written out of the novel in later editions as he lost favor. According to an article I unexpectedly found in some classical guitar literature, this Seguin was in fact a real person, and was Irish (?!). (I suppose a French guy could have moved to Ireland, married, and had a kid.) He later went to America and got involved with a Native American tribe.

]]>Here’s Harold Bloom on that last paragraph (hopefully the link will work): https://books.google.com/books?id=GjIlriah_2EC&pg=PA150 I think modern practice would be to leave out the final period.

]]>My understanding was that the crown of the hat is meant to go in front of your hand. For example Fig. 7 shows the protective pad inside the crown, and this is where the ball is supposed to end up.

Either way, it doesn’t strike me as very practical: it seems that the bill of the hat would make it very awkward to get your hand inside the glove in time to make the catch.

Among the (many!) related ideas, I think that https://patents.google.com/patent/US20070113321A1

has the most practical implementation. It doesn’t look great, though…

When I was a student in the early 1970s, I was dragged to the Plaza to try the burger. I don’t recall being especially impressed. My most memorable burger was one with finely chopped onion kneaded throughout the hamburger and cooked slowly over a grill. I haven’t had one since I was a kid.

Squash blends nicely with Southeast Asian curries and is quite yummy. Now most of the Thai/Lao restaurants in my area have versions of this. I fear it will be a little while though before I can return to Lao Laan-Xang. Earlier this year I had one of their stir fries, and several hours later I was projectile vomiting in the bathroom. It sometimes takes a long while for the vomit association to dissipate. It was probably a bug already brewing in my stomach and not their fault at all. I’ve had a lot of good food from Lao Laan-Xang over the years.

]]>2) We don’t know! And of course any one of my opinions — about morals or anything else — might be wrong. It might not even take 100 years to figure it out! As Quine said, I always think I’m right, but I don’t think I’m always right.

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