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Jul
18
awarded  Nice Question
Jul
13
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 $f$.
Jul
13
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.
Jul
13
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."
Jul
12
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 $\lnot A$.
Jul
12
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.
Jul
12
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.
Jul
12
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.
Jul
12
answered Vector spaces - $\mathbb{R}^n$ and $\mathbb{R}^m$
Jul
12
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).
Jul
12
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.
Jul
12
comment Limit of ratio of sequences
Your first limit after "I proceeded" does not need to exist. For example, take $a_n=n^2$, $b_n=n^3$, $s=0$.
Jul
12
comment Does Pi contain all possible number combinations?
How do you know that the set of non-normal numbers is Lebesgue measurable?
Jul
12
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.
Jul
12
revised Probability of winning a tournament by winning all matches in tournament
added 7 characters in body
Jul
12
answered Probability of winning a tournament by winning all matches in tournament
Jul
12
reviewed Approve Prove $\{a(x,y,z)=(ax,y,z)\}$ is a vector space
Jul
12
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.
Jul
12
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.
Jul
11
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 $1$.