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What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?

Motivation In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
Max Lonysa Muller's user avatar
0 votes
0 answers
52 views

Classifier-specific lower bounds on the misclassification rate in binary classification

Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
tim523's user avatar
  • 13
4 votes
1 answer
258 views

Bounds for the crossing number in terms of the braid index?

Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$? For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
Charles's user avatar
  • 9,114
1 vote
0 answers
32 views

Are there known bounds on these ratios of chromatic polynomials?

The chromatic polynomial $P(G, \lambda)$ gives the number of proper vertex colorings of the graph $G$ with $\lambda$ colors. I'm interested in how many possible colorings you loose when you add an ...
M. Stern's user avatar
  • 111
0 votes
1 answer
279 views

Lower-bound on smallest singular-value of rectangular random matrix

Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
dohmatob's user avatar
  • 6,853
8 votes
2 answers
1k views

Lower bound on exponential sums

Let $k\geq 2$. Consider the following norm of exponenetial sum: $$ I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy. $$ Bourgain mentioned on Page 118 of https://...
Thomas Yang's user avatar
1 vote
2 answers
1k views

Bounds on the smallest real positive root of a polynomial

I'm trying to find upper and lower bounds of the smallest positive root of a polynomial, stated in terms of its coefficients. As I appreciate it might be a very general problem, My specific interest ...
Amir Sagiv's user avatar
  • 3,574