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1d
revised Spectral radius and dense subspace
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1d
comment Parameterization which is closed under addition
I don't really understand what you are looking for. What does "similarly re-parameterized" mean for you? (If this is part of a bigger problem, it could be a good idea to say something about it).
1d
comment Prove norm inequality
Are you sure that $\lVert x \rVert^2=x_1^2-x_2^2$? I think there should be a $+$ sign there.
1d
awarded  Nice Answer
2d
comment Spectral radius and dense subspace
@hal: you should un-accept this answer at once, as it is wrong.
2d
comment Spectral radius and dense subspace
@RobertIsrael: of course, you are right, it must be continuous. So this example is flawed and cannot be repaired, as you show in your edited answer (which I have just seen, actually)
2d
answered Spectral radius and dense subspace
2d
comment Is the empty set an open ball in a metric space?
Well, it can be considered as the ball of radius $0$ I guess. Is it that important?
Feb
3
comment proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$
The invariance of domain theorem relies on Brouwer's fixed point theorem, whose proof relies on homology or cohomology, AFAIK. There lies the hidden calculation you mention.
Feb
3
answered proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$
Feb
3
comment condition number of matrix plus constant times identity
Didn't see your comment. I agree. This must be true for normal matrices (i.e., $A^TA=AA^T$), which include symmetric ones. P.S. Yes, it is true. If $A$ is normal then $A=U\rm{diag}(\lambda_j)U^\star$, where ${}^\star$ means "conjugate and transpose" and $U$ is unitary. So in particular $A^T=U\rm{diag}(\overline{\lambda_j})U^\star$ and one has that $A^TA$ has $|\lambda_j|^2$ as eigenvalues. HTH
Feb
3
comment condition number of matrix plus constant times identity
This might be true for normal matrices, however.
Feb
3
comment condition number of matrix plus constant times identity
I was ready to bet that the eigenvalues of $A^TA$ were $|\lambda_j|^2$, $\{\lambda_j\}$ being the eigenvalues of $A$, but unfortunately I realized it is not true. The matrix $A=\begin{bmatrix} 0 &1 \\ 0 & 0 \end{bmatrix}$ only has $0$ as an eigenvalue but $A^TA=\begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix}$ has the unexpected eigenvalue $1$.
Feb
2
comment Showing that a function is uniformly continuous but not Lipschitz
HINT: The inequality you want to disprove can be rephrased as follows: "the function $|x|^{-\frac12}$ is bounded on $(0,1]$". Is this true? If you can show it's not, you are finished.
Feb
2
answered Subset of $(l^{2},d_{2})$ is open
Feb
2
comment non homogeneous heat equation?
HINT: Look for the keywords "Duhamel's formula for the heat equation".
Feb
2
comment Showing interior of a set is empty
... its graph lies in the $\delta$-ball centered at $f$? I claim you can. Since $\delta$ is arbitrary, this shows that $A$ has empty interior.
Feb
2
comment Showing interior of a set is empty
That metric space is one of the few functional spaces that can actually be drawn. Represent a continuous function $f$ by its graph. Then the open ball of radius $\delta$ around $f$ consist of all continuous functions whose graph lies in the $\delta$-tube around the graph of $f$. It is more easily pictured than said. Now the set $A$ of this posts consists of functions that satisfy a Lipschitz condition around the point $x_0$. Picture one such function and fix $\delta>0$. Can you find a non-Lipschitz continuous function, such as $\sqrt{|x-x_0|}$, such that ...
Feb
1
comment Reference request: about inverse Laplacian operator
@tankonetoone: Yes, it does. Look in the chapter about Hilbert spaces if you want (especially around something called "Hilbert basis"). Or follow the reference of TrialAndError.
Feb
1
comment Uniform convergence, wrong answer?
The "master solution" used a trick to compute $\sup_{x\in[0,1]} |x^n(1-x)^n|$ known as "completing the square". You are not forced to use such tricks, however. You could equally well use calculus to compute that sup. Compute the derivative of $x(1-x)$, see where it vanishes, conclude it is a maximum, etc etc...