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seen Jul 17 '11 at 17:34

Apr
10
awarded  Yearling
Apr
10
awarded  Yearling
Jan
11
awarded  Nice Answer
Apr
10
awarded  Yearling
Jul
17
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.
Jul
17
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 $B$: $(0),(1),(2),(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(0,0,1),(0,1,0),(0,‌​1,1),(1,0,0),(1,0,1),(1,1,0)$.
Jul
17
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.
Jul
17
revised Trace of sequence of natural numbers
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Jul
17
comment Trace of sequence of natural numbers
@Brian: thanks! I've tried to fix things up as you suggested.
Jul
17
revised Trace of sequence of natural numbers
added 304 characters in body
Jul
17
revised Trace of sequence of natural numbers
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Jul
16
answered Trace of sequence of natural numbers
Jul
12
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 $1$.
Jul
12
revised Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$
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Jul
12
answered Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$
Jul
7
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 $0<k<1$ then I guess $0.5^t=k$, so $t=-\log_2 k$.
Jul
6
awarded  Altruist
Jul
5
comment How to show determinant of a specific matrix is nonnegative
Nice! @leonboy: I think the $\theta_i$ can be arbitrary, provided they sum to $\pi$ (since this is what makes $z_{N+1}=-z_1$ work).
Jul
2
answered Adding and Multiplying Polynomials Recursively
Jul
2
comment How to determine whether to find an upper bound or lower bound
Isn't your question a tautology then? To prove that it's bounded above, bound it from above; to prove that it's not bounded above, bound it from below (by some sequence you know to not be bounded above). Similarly if you want to prove/disprove that it's bounded below.