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 Jul18 awarded Nice Question Jul13 comment Is there a simple sufficient condition for a function to depend “only on $r$”? A necessary (but not sufficient) condition for dependence only on $r$ is that you must get the same function if you exchange $x$ and $y$. That simple test would already have identified the student's answer as wrong. More generally, if $g$ has any symmetry, that symmetry must also show up in $f$. Jul13 comment Typesetting the mathematical expression “const.” The reason could of course just be that if you just type const. without any further commands in $\rm\LaTeX$ formulas, you get italics. Jul13 comment When a function is a dimension? @JonathanHebert: So what would you then call a criterion in the everyday-language sense that is not necessary (for example, a sufficient criterion)? The "being seven" criterion sounds ridiculous because it's ridiculously narrow. More similar to my dimension sentence would be "an obvious criterion for the term "numbers" making sense for elements of an algebraic structure is if that structure includes the integers." Jul12 comment Logically equivalent formulas and contradiction @Sushil: $A\implies(\lnot B\land B)$ is not a contradiction, it is the statement that assuming $A$ would imply a contradiction. As such, it is equivalent to $\lnot A$. Jul12 comment When a function is a dimension? I didn't claim it's always true. I said it's one obvious criterion. Which implies that there are others as well. Jul12 comment When a function is a dimension? One obvious criterion would be that if you can apply the definition to mathematical objects where you already have the term "dimension" defined, and it turns out that for those objects it gives exactly the previously defined dimension, then it probably makes sense to call that newly defined quantity a dimension as well. Jul12 comment Can a vector field be conservative if its domain is not a star domain? Another way to get a conservative function on a non-star shaped domain is to take a conservative function on a star-shaped domain and restrict it to a non-star shaped subset of that domain. Jul12 answered Vector spaces - $\mathbb{R}^n$ and $\mathbb{R}^m$ Jul12 comment Order in writing composed morphisms @darijgrinberg: But that's exactly what we do with operator notation on linear spaces: $Av$ is the application of the linear operator $M$ to the vector $v$ (in finite dimension: multiplication of the matrix M with the vector v), while $AB$ is the composition of linear functions (matrix multiplication). The point is, you have to know what sort of object you are dealing with (another example: if $\alpha$ and $\beta$ are scalars and $v$ is a vector, then $(\alpha\beta)v$ also involves two different operations; one field multiplication, and one scalar multiplication of a vector). Jul12 comment Order in writing composed morphisms Another argument for the "reverse" (compared to the traditional) notation is that it gives a more natural composition rule for relations: With the usual rules, you get $a(R_1\circ R_2)b \iff \exists c: cR_1b\land aR_2c$. It would be much more natural to define it as $a(R_1\circ R_2)b \iff \exists c: aR_1c\land cR_2b$. But with the usual composition rule, that would not reduce to the usual function composition. It would, instead, reduce to the reverse function composition. Jul12 comment Limit of ratio of sequences Your first limit after "I proceeded" does not need to exist. For example, take $a_n=n^2$, $b_n=n^3$, $s=0$. Jul12 comment Does Pi contain all possible number combinations? How do you know that the set of non-normal numbers is Lebesgue measurable? Jul12 comment Probability of winning a tournament by winning all matches in tournament Your terminology is very confusing: You are using "tournament" for two different things, a combination of three matches, and a series of tournaments in the first sense. In particular, your players can not win the tournament, and yet win the tournament … because the "tournaments" in both parts of the sentence are different things. Jul12 revised Probability of winning a tournament by winning all matches in tournament added 7 characters in body Jul12 answered Probability of winning a tournament by winning all matches in tournament Jul12 reviewed Approve Prove $\{a(x,y,z)=(ax,y,z)\}$ is a vector space Jul12 comment Computing the volume of this weird object, @mathcounterexamples.net: in the interval $[-1,1]$ the sphere of radius $2$ has cylindric radius $\ge\sqrt{3} > 1$. Since the function is restricted to be $\le 1$, the surface clearly is completely inside the sphere. Jul12 comment Show that $\frac{1\cdot 3\cdot 5\cdot \ldots \cdot (2n-1)}{1\cdot 2\cdot 3\cdot \ldots \cdot n}\leq 2^{n}$ for all $n\in\mathbb{N}$. If you really feel that you have to prove that $2^k$ is positive, you can of course do that using induction as well. It definitely is not impossible. Jul11 comment Zero to the zero power - Is $0^0=1$? $0^0=999$ would be a contradiction to the power laws, because then $(0^0)^2 = 999^2 \ne 0^{0\cdot2} = 999$. The only two values for $0^0$ consistent with the power laws are $0$ and $1$.