Jonas
Reputation
700
Top tag
Next privilege 1,000 Rep.
Create new tags
 Mar 4 comment Is this a correct proof for $\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}=0$? I believe you mean $y = r \sin \theta$ Feb 27 comment On the justification behind a big O manipulation (quick question). Stirling's approximation is likely useful here: $n^n \simeq n!/(\sqrt{n})$ Feb 14 awarded Civic Duty Jan 29 accepted Vector perpendicular to timelike vector must be spacelike? Oct 27 accepted Norm of operator between two $L^p$ spaces? Oct 24 revised Real Analysis, Folland Proposition 2.1 fix typo in countable union markup Oct 24 answered Real Analysis Folland, Corollary 2.2 Oct 23 answered Real Analysis, Folland Proposition 2.1 Sep 20 awarded Nice Question Sep 20 awarded Yearling Aug 20 awarded Popular Question Apr 17 comment Contractions and finding Fixed Points Feel free to read more about fixed points here. I am sensing some confusion here about the fixed point; note that $D$ is an operator defined on the function space $C[0,1]$, not on the interval $[0,1]$. Therefore it acts on functions $f$, and so its fixed point is a function, not a point in the interval $[0,1]$. Apr 17 comment Contractions and finding Fixed Points By definition, a fixed point $f$ of the contraction satisfies $D(f) = f$. Apply the definition of $D$ to figure out what $f$ must be. For example, since $Df(x) = x - 2/3$ on $[2/3,1]$, we must have that $f(x) = x - 2/3$ on the same interval also. Apr 17 comment Contractions and finding Fixed Points Well, it looks like my first comment has the gist of it. Just note that on $[2/3,1]$, $Df$ and $Dg$ are equal, hence $|Df(x)-Dg(x)| = 0$, which finishes the proof that $D$ is a contraction. You have a sign error in your last equation, hence the confusion. Apr 17 comment Contractions and finding Fixed Points Please clean up the fractions in your post; it's very difficult to understand right now. Assuming that your post accurately defines $D(f)$ on the last third of $[0,1]$, then $D(f)$ and $D(g)$ are in fact equal on that interval, and so their difference on $[2/3,1]$ will not contribute anything towards the sup norm $||D(f)-D(g)||_\infty$. Mar 15 comment Floor Function Homomorphism and Isomorphism The group operation here is addition, not multiplication. Mar 12 awarded Popular Question Mar 12 comment Show that $\lambda$=1 as eigenvalue, find one corresponding eigenvector Your matrix multiplication is incorrect; for example, the first row of your last set of equations should be $5v_1 - 2v_2 + 3v_3 = 0$ Feb 14 comment “Length” of an element in a free group You are looking for the word metric on a group, I believe. Feb 12 comment Apparent counterexample to the Picard-Lindelöf theorem What are the assumptions of the Picard-Lindelof theorem? Does $xy^{1/2}$ satisfy those assumptions?