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Mar
30
comment Is $\mathbb{C}-\{0\}$ homeomorphic to $\mathbb{R}$?
In this case, I think it's “trained minds think alike”. This argument is fairly standard, though perhaps not the first thing you think of if you haven't seen it before.
Mar
30
answered Why is $\lim _{n \to \infty} n \arctan(\frac{x}{n})=x?$
Mar
30
answered Derivation of the polarization identities?
Mar
30
reviewed Approve Does the following infinite series converge?? (Leibnitz)
Mar
28
comment positive linear functionals are bounded in $C^*$-algebras
@Freeze_S With notation as in the answer, $b=\frac12(a+a^*)$, so $\|b\|\le\|a\|$ by the triangle inequality. Similarly, $\|c\|\le\|a\|$. So $|f(a)|^2=|f(b)+if(c)|^2=f(b)^2+f(c)^2\le C^2(\|b\|^2+\|c\|^2)\le 2C^2\|a\|^2$, where $C=f(e)$.
Mar
27
comment Show that if $w^3=1$, then $1+w+w^2=0$
Yes, but it should have been stated explicitly.
Mar
27
comment Correct typography for using Leibniz Notation
How can one possibly answer a question like the $\iota$ one? There are a lot of letters that could be used, but aren't. Yes, it's an ISO standard, though when I last saw it years and years ago, it was as an NS (Norwegian Standard, which follows the ISO standards closely).
Mar
27
comment Difference between $\ell ^{\infty}$ and $\ell^p$
As @Wouter said. However, note that this is for finite sequences. Still, it serves to motivate the definition of $\ell^\infty$ even for infinite sequences (just replace the maximum by a supremum).
Mar
27
comment Correct typography for using Leibniz Notation
The vast majority of mathematics journals and books use italic $d$, I think. But there exists an international standard for notation in technical and engineering fields that specifies the upright $\mathrm{d}$ for this purpose, and similarly for $\mathrm{i}=\sqrt{-1}$ and $\mathrm{e}=2.71828…$ (sorry, I don't have the reference on hand).
Mar
25
comment For differentiable functions $f,g$, $\nabla f(x)=g(x)x$. Then $f$ is constant on S.
To add some intuition to what @AndrewD.Hwang said: The assumption says that $\nabla f$ points in the radial direction, which means that $f$ will not change when you move along a sphere centered at the origin. So $f$ should be a function of $\|x\|$ alone.
Mar
20
comment series divergence $\sum_{k=1}^{\infty}\sqrt{\tan^{-1}(\frac{1}{k^2})}$
Actually, if you examine the proof of the validity of the limit comparison test, you will see that it does produce a series of smaller terms to compare with.
Mar
19
comment Is there a definition of ${\forall}$ that doesn't use the concept of propositional function?
A similar comment applies to the concept of a propositional function, which is usually considered a meta concept; i.e., it is not a concept within the theory under study, but rather a concept about this theory.
Mar
19
comment Is there a definition of ${\forall}$ that doesn't use the concept of propositional function?
I have never come across a formal system in which $\forall$ is subject to definition. In formal first-order logic, the symbol is just there, and has to satisfy the axioms of first-order logic. There is, however, a definition of the interpretation of $\forall$ in a model in first order logic, and that definition resembles the one you gave. However, note that this definition is not a definition inside the first-order logic, so there is no circularity involved.
Feb
25
revised How to write Kelvin equation in a different way
Fix some LaTeX
Feb
25
comment How to write Kelvin equation in a different way
You are aware that $\ln(\mathrm{e}^x)=x$, right? (Some people like to write $\mathrm{e}=2.71828…$ with an upright e. When you have other $e$s in the same context, this becomes essential.)
Feb
11
comment Triangles - sin, cos etc.
@Nehorai In many parts of the world (e.g., much of Europe) one uses the comma instead of a period as decimal separator.
Feb
11
comment What is the sum of all the natural numbers between $500$ and $1000$.
Looks like a good one to emulate, yes. How about: Find the sum of all even numbers in the given range. Find the sum of all multiples of 14 in the given range. Subtract.
Feb
5
answered How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?
Feb
2
comment Idea behind the tangential vector space?
To expand a little bit on the comment by @HagenvonEitzen, try to replace $U$ by the unit sphere in $\mathbb{R}^n$. You will find that the tangent space at any point is $n-1$-dimensional, and can in fact be identified with the hyperplane touching the sphere at the given point. And yet, there is no sensible way to identify all these spaces with $\mathbb{R}^{n-1}$ in a coherent way.
Jan
31
comment Banach Tarski proof understanding
Probably, assuming that $S$ being decomposable into the elements of $X$ means that you can write $S=P_1\cup\cdots\cup P_n$ (with the $P_i$ pairwise disjoint) and $P_i$ congruent to $Q_i$ for $i=1,\ldots,n$.