Dave L. Renfro
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 2d comment When do we write a superscript before a smbol? To add to @Rob Arthan's comment (I've been away and didn't see your reply to my comment until just now), in most other areas of pure mathematics $S^T$ represents the set of functions from $T$ into $S.$ But as Rob pointed out, this can be ambiguous when working with cardinal and ordinal numbers (and these "numbers" represent certain sets). 2d comment When do we write a superscript before a smbol? It's used in set theory, for example $^{\omega}{\omega}$ represents the set of all functions from $\omega$ (the positive integers) into ${\omega}.$ Also in denoting the tetration operation. 2d comment Can we define the derivative of a function in arbitrary metric space in the following way? Also of possible interest: Generalized Ordinary Differential Equations in Metric Spaces by Bøetislav Skovajsa (2014 Ph.D. Dissertation) 2d comment Can we define the derivative of a function in arbitrary metric space in the following way? Of possible interest: Metric derived numbers and continuous metric differentiability via homeomorphisms by Jakub Duda and Olga Maleva (2006); On relations among metric derived numbers by Martin Koc (2009); Differential equations in metric spaces by Jacek Tabor (2002) 2d revised Question concerning limit superior and inferior. My original definition of $b_n$ in #2 only gives us $\limsup a_n \geq \liminf b_n$ Feb 3 answered Question concerning limit superior and inferior. Feb 3 comment Why do people prefer cosine to sine when speaking of harmonic oscillation? Interesting. I would have guessed $\sin,$ but I don't have any books around me to look at right now. The reason I would guess $\sin$ is that the approximations $\sin x \sim x$ and $\sin x \sim x - \frac{1}{6}x^3$ are used quite a bit in physics and engineering (e.g. diff. eq. for a pendulum when you want to avoid elliptic functions), more so it seems to me than the corresponding approximations for $\cos.$ Feb 3 comment “Peak lemma” and explicit monotone subsequence @Gerry Myerson: (in case you're interested) A stronger version of the "peak lemma" (first time I've heard this term, by the way) can be proved using Ramsey theory. See this 4 August 2009 sci.math post. Feb 3 comment Complex Root of Unity Analogue of Forward Difference Operator You might find something relevant in one of these two books, or others like them published in the mid-19th century: The Calculus of Operations by John Paterson (1850) AND A Treatise on the Calculus of Operations by Robert Carmichael (1855) Feb 2 comment Law of Excluded Middle Controversy Your 2nd paragraph got me wondering whether this equality issue still arises if we restrict ourselves to comparing two rational numbers each known to have a finite decimal expansion, and the rational numbers are presented to us in that form. Or even whether this issue arises when comparing two positive integers given in decimal form. Note that no matter how many digits we've checked and found to agree for the two positive integers, we wouldn't know whether the integers are different because we don't know when the digits will end. Maybe not as problematic, but I can still see issues with this. Feb 2 comment Declaring function range, what else can be excluded than values where it's undefined? +1 for nice completion of the square argument! For what it's worth, I notice this function is half of $x + \frac{1}{x},$ and I believe I've seen several methods for showing $x + \frac{1}{x} \geq 2$ (at least when $x>0),$ but I don't remember them now (except by differentiation, and now of course after seeing your solution, completion of the square). Feb 2 comment How can revolving an infinite area have a finite volume @Yuriy S: I've also wondered about this for many years (decades, in fact). For some reason the one-and-two dimensional version seems to cause MUCH less concern than the two-and-three dimensional version. Feb 2 comment How do I solve the following equation: $z^4+z^3+z+1=0$ Related: The roots of $t^5+1$. Note that $z^5 - 1 = (z-1)(z^4 + z^3 + z^2 + z + 1)$ and $z^5 + 1 = (z+1)(z^4 - z^3 + z^2 - z + 1).$ Feb 2 comment Angle trisection existence "Yes" to your question. It's analogous to $\sqrt{2}$ being impossible to create using integers and the four arithmetic operations, or $\frac{1}{2}$ being impossible to create using integers and the arithmetic operations excluding division, or being impossible to create the antiderivative of $e^{-{x}^{2}}$ using precalculus functions, or being impossible to create the trig. functions using polynomials and the four arithmetic operations and composition of functions, or being impossible to create a nonzero number using only $0$ and the arithmetic operations excluding division, etc. Feb 2 revised A question on set C[0,1] Yikes! I accidentally misstated what the 2nd half proves! Feb 2 comment Find the set of all accumulation points for the following set You may want to say that $i = \sqrt{-1},$ since the letter $i$ is often used as a variable having positive integer values (e.g. summations, indexing a collection of sets, etc.) and thus some people might think you overlooked including $i$ in the condition $m,n \in N.$ Feb 2 revised A question on set C[0,1] Making the 2nd half more logically sound Feb 2 answered A question on set C[0,1] Feb 1 awarded Nice Answer Feb 1 comment Lebesgue measure has the Darboux property For those interested in additional aspects about this property, see this 25 November 2005 sci.math post archived at Math Forum. Also, google the phrase "range of a measure".