Robert Israel
Reputation
26,180
96/100 score
 15h revised Finding functional equations that a given function satisfies added 649 characters in body 1d comment Rational power Napier number The six exponentials theorem implies that there are at most two primes $p$ such that $p^e$ is algebraic. 1d comment Style guide for referring to past work @D.R. Of course past tense, unless you're psychic ("Roe will prove this in 2018"). 1d comment Average minimum number of random k-sparse vectors in GF(2) to span the whole space? You might look at this related MO question of mine and its answer by Kevin Costello. 1d answered Finding functional equations that a given function satisfies 2d answered Logarithms of matrices in the disk-algebra Apr 28 comment Banach-Mazur distance between the cube and the octahedron $n=5$ also seems difficult for the Global Optimization Toolbox. Apr 28 revised Banach-Mazur distance between the cube and the octahedron added 22 characters in body Apr 28 revised Banach-Mazur distance between the cube and the octahedron added 22 characters in body Apr 28 answered Banach-Mazur distance between the cube and the octahedron Apr 28 comment Extremal Lipschitz convex functions You mean Lipschitz-$1$, i.e. $|f(x) - f(y)| \le \|x-y\|$? Apr 27 comment Can the matrix exponential be equal to the elementwise exponential $\pmatrix{\ln(-4/3) & \ln(-2) & \ln(-2)\cr \ln(-2) & \ln(-4/3) & \ln(-2)\cr \ln(-2) & \ln(-2) & \ln(-4/3)}$ Apr 27 comment convergence of ODE Phase plane analysis. Apr 25 comment Mathematicians with Aphantasia (Inability to Visualize Things in One's Mind) I don't think I have aphantasia, but I don't generally produce very detailed and vivid mental images. I think this can be an advantage when dealing with mathematical objects: their geometrical relationships, when they have them, tend to be rather simple, and you don't want to get distracted by unnecessary details. Also, most of mathematics is dealing with objects, e.g. those of more than three dimensions or with nontrivial topology, which nobody can really visualize. It helps to be used to not panicking when something resists visualization. Apr 25 comment Adapting arguments and plagiarism Of course it would be better if you could come up with a brilliant new method that blows all these problems away. In the absence of that: most research makes incremental progress by adapting existing tools to a slightly different situation. Apr 22 answered On number of perfect matchings Apr 22 comment How to estimate a specific infinite matrix sum Note that $x^T M x = \|x\|^2$ if $e \cdot x = 0$. Let's say $n$ is even. For any $y \in \mathbb Z^{n/2}$, the vector $x = (y_1,-y_1,y_2,-y_2,\ldots,y_{n/2},-y_{n/2})$ has $x^T M x = \|x\|^2 = 2 \|y\|^2$. Thus for any $k$, $$S_{M,k} \ge \sum_{y \in \mathbb Z^{n/2}} e^{-2 \|y\|^2} = \left(\sum_{y \in \mathbb Z} e^{-2 y^2}\right)^{n/2}$$ which does grow exponentially in $n$. Apr 21 comment Matrix diagonalization after a completely positive transformation Of course: from $U$ and $D$ you compute $A$, with that and the $L_j$ you get $\widetilde{A}$, and from that you get (non-uniquely, of course) $\widetilde{D}$ and $\widetilde{U}$. But if you want a much simpler way than that, it's rather unlikely. Apr 20 revised Spectral radius's relation with row sum edited body Apr 20 answered Spectral radius's relation with row sum