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 Apr10 awarded Yearling Apr10 awarded Yearling Apr10 awarded Yearling Jan11 awarded Nice Answer Apr10 awarded Yearling Jul17 comment Trace of sequence of natural numbers The conjecture I made above is correct, as can be seen by considering the tree of preimages of $(1,2)$ under $t$ (the children of a node form the set of preimages of that node, which is the set of permutations of any one preimage). There is one infinite path given by cycling through $R$, but every other path in the tree is finite since it ends in a node of the form $(c_0,\dots,c_{m-1})$ where $c_{m-1}=0$, and these have no children. The nodes of the tree are the sequences I conjectured. Jul17 comment Trace of sequence of natural numbers I would conjecture that the sequences of type $R$ are the following, and all the others are type $B$: $(0),(1),(2),(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(0,0,1),(0,1,0),(0,‌​1,1),(1,0,0),(1,0,1),(1,1,0)$. Jul17 comment Compactness of Multiplication Operator on $L^2$ @paul: I don't know what you mean. There are certainly finite rank operators on $L^2[a,b]$; for example, the map sending $f$ to the constant function $\int f\,d\mu$ has rank one, and is of the form $\langle\cdot,1\rangle 1$; more generally, if $g,h\in L^2[a,b]$ then $\langle\cdot,g\rangle h$ has rank one. Jul17 revised Trace of sequence of natural numbers deleted 20 characters in body Jul17 comment Trace of sequence of natural numbers @Brian: thanks! I've tried to fix things up as you suggested. Jul17 revised Trace of sequence of natural numbers added 304 characters in body Jul17 revised Trace of sequence of natural numbers deleted 21 characters in body Jul16 answered Trace of sequence of natural numbers Jul12 comment Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$ @Sunni: sorry, I don't know any references for this sort of question. But the eigenvalue comparison doesn't seem to work out: if you take $t=1$ above, then $1/\sqrt2$ is the only eigenvalue of $|A|\circ |B|$ (with multiplicity $2$), but the eigenvalues of $|A\circ B|$ are $0$ and $1$. Jul12 revised Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$ deleted 14 characters in body Jul12 answered Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$ Jul7 comment creating smooth curves with f(0) = 0 and f(1) = 1 How about $f(x)=x^t$ where $t$ is a positive real constant? This has $f(0)=0$ and $f(1)=1$, and if $f(0.5)=k$ where \$0