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Apr
8
comment What does this notation mean? $x \mapsto f(x)$
@YoTengoUnLCD: if you consider it tautologous, fair enough. (But there are people who would write their functions as $x\mapsto xf$ or $x\mapsto x^f$ or similar. Not stating this can lead to genuine confusion - does $fg$ mean "do $f$ then $g$" or "do $g$ then $f$"? Writing explicitly that you will denote the image of $x$ under $f$ by $f(x)$ avoids this potential confusion. Also, for instance, with more arguments, you might write $g(x,y)$, but would prefer to write $x*y$ rather than $*(x,y)$; writing this out explicitly avoids confusing your reader.)
Apr
8
comment Find all Gaussian integers $α, β, γ$ such that $αβγ = α + β + γ = 1$
"If any of α,β,γ are equal from that list they cannot add to 1" - apart from the case (1,1,-1) which is easily checked to fail.
Mar
27
reviewed Approve Let f(x),g(x) be complex polynomials, if f(x) | g(x) and g(x) | f(x) then..
Mar
27
comment Does a terminating recurrence relation diverge?
As you've noticed, the expression $u_{k+1} = \frac{4}{u_k + 2}$ doesn't make sense when $k = 6$ (so there should really be a caveat in the definition). What is your definition of 'sequence', and does $\{u_k\}$ fit it? If this isn't even a sequence (in this case, because it fails to be infinite), it doesn't make sense to ask whether or not it converges.
Mar
27
comment Can I make an assumption about arbitrary numbers in a proof?
No, you may not assume anything that you can't prove (or your lecturer / teacher hasn't proved). Can you prove that $Q$ exists? (Hint: try to find $P_1$ with $P_1(x_1) = y_1$ and $P_2$ with $P_2(x_2) = y_2$, and combine them somehow to find $P$ with $P(x_1) = y_1$ and $P(x_2) = y_2$.)
Mar
27
reviewed Reject What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?
Mar
27
reviewed Approve Expressing an integral in closed form
Mar
27
comment Let f(x),g(x) be complex polynomials, if f(x) | g(x) and g(x) | f(x) then..
@MattSamuel Depends on your definition of "divides", surely? In a commutative ring (here $\mathbb{C}[x]$) the natural definition seems to me to be "$g|h$ iff there exists $f\neq 0$ with $fg = h$". If you want to make an exception for $0$ somewhere, go ahead, but definitions vary from author to author, so a discussion of which is correct doesn't seem all that useful. I just flagged it up as a potential issue.
Mar
27
answered Let f(x),g(x) be complex polynomials, if f(x) | g(x) and g(x) | f(x) then..
Mar
27
reviewed Reject help me understand this given solution from gelfand algebra text.
Jan
16
comment Why are primes considered to be the “building blocks” of the integers?
@zod Edited, thanks!
Jan
16
revised Why are primes considered to be the “building blocks” of the integers?
edited body
Jan
13
awarded  Nice Answer
Jan
11
comment $X^2$irreducible but not prime
(Are $X$ and $x$ the same thing, and do you mean $X^i$ instead of $X_i$?) $X^2$ certainly looks prime here to me.
Jan
11
answered Reverse Product Rule ODE
Jan
11
reviewed Approve $2\times 2$ matrices forming the Klein $4$-group.
Jan
11
revised Why are primes considered to be the “building blocks” of the integers?
added 44 characters in body
Jan
11
answered Why are primes considered to be the “building blocks” of the integers?
Jan
11
comment Group Duality with respect to Generators and Relations
(2/2) On the other hand, there certainly are some notions of duality that arose from mathematical questions, which you may be interested in looking up. A simple one would be the concept of a dual vector space (vector spaces are, of course, continuous groups too), which arises by considering functions on that space (or matrix transposition). A harder one: Pontryagin duality comes from trying to do Fourier analysis on certain groups, and crops up very naturally in a bunch of places, including e.g. the character theory of finite abelian groups.
Jan
11
comment Group Duality with respect to Generators and Relations
(1/2) I think my real point is: in order for such a definition to be mathematically meaningful, its conception needs to be mathematically well-motivated. You can't just say "I want duals to exist; where are they?". Better questions: what do you want $G^D$ to do? Or how do you want it to look relative to $G$? I see no mathematical reason to swap the generators and relations of a group (and no guarantee that it could be made to work). My geometric proposal was just something slightly more well-motivated (but, if you work through the details, you'll see it's also not very interesting).