7,961 reputation
1642
bio website people.bath.ac.uk/mdp33
location Bath, United Kingdom
age 24
visits member for 2 years, 4 months
seen 22 mins ago

Postgraduate student at the University of Bath, UK, studying geometry and representation theory. Currently thinking about cluster algebras and related objects.


Jul
11
comment If $A$ is compact and connected, then is $\Bbb R^2\setminus A$ connected?
It's not really a stupid question! But the lesson should probably be to always try out some examples first - Heine-Borel makes it fairly easy to generate lots of them. (Of course, if you didn't have a good handle on what it means to be compact, then generating example might be more problematic, but it seems that you do.)
Jul
11
comment How Max plus algebra is different from conventional algebra?
In a sense, your tag provides an answer to the question - the max-plus algebra gives us a neat way to tropicalize a variety, and then there are various theorems relating properties of the original variety to properties of the tropical variety, which are potentially easier to compute because of the combinatorial nature of the object. I believe there are other applications of max-plus not directly to do with tropical geometry, but I don't know what they are.
Jul
10
comment writing papers: definition in word or formula?
This question might be too opinion-based. I don't have any real preference between the three options. If the set had a more complicated definition, then I would probably be more strongly against C) and more strongly in favour of A). The aim should always be clarity, but this can be extremely subjective.
Jun
26
comment Origin in homogeneous coordinates
@HaraldHanche-Olsen Standard that may be, but canonical it is not! ;-)
Jun
26
comment Origin in homogeneous coordinates
To elaborate a little - the origin of a vector space is special because it is the additive identity under the group structure. But once you projectivise the vector space, the group structure disappears - and there is no "special" line in the vector space to play the role of the origin in projective space.
Jun
26
comment Origin in homogeneous coordinates
There is no origin in projective space!
Jun
26
comment Building quotient rings
@o2genum I'm not sure this is a concrete question - you already have described the rings! If you want to describe them in another form, you need to be quite specific about what that form should be. Asking whether the first two are isomorphic to $\mathbb{Z}_n$ is a perfectly good question, but "what ring is $\mathbb{Z}[\sqrt{-2}]/(2)$ isomorphic to?" is too open-ended (unless there's a "standard" ring it's isomorphic to that I don't immediately see).
Jun
19
comment Is it true that every eigenvalue has at least one eigenvector?
You could in theory define an eigenvalue to be a root of the characteristic polynomial of $f$. Michael's answer provides half of the proof that these two definitions are equivalent.
Jun
17
comment Is the intersection of dense sets dense?
@freebird The complement of a nowhere dense set is always dense - indeed, one definition of a nowhere dense is that the complement of its closure is dense. But the complement of a dense set is not necessarily nowhere dense, as in the example in your question. So you while you can't prove that the intersection of two arbitrary dense sets is dense (because this is false!) the sets $A'$ and $B'$ in your setup have the stronger property that their complements $A$ and $B$ are nowhere dense. I agree with Ittay that this is probably not the best way to think about the problem though.
Jun
5
comment Solving matrices
I don't think I entirely understand your question - but I will try to say something useful. The question you link to asks you to calculate the determinant of the matrix, which you can do without thinking at all about a linear system that the matrix might represent.
Jun
4
comment Questions for first year students at the University.
Some pedantry - I would argue that $1+2+3+\dotsb+100$ already is written as a sum. Maybe you mean "using $\Sigma$-notation" or something like that?
May
29
comment Indecomposable quiver representations
Literally, no, because there are several representations with the same dimension - for example, all the vertex simple representations have dimension $1$. But maybe you mean "dimension vector"?
May
20
comment Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent.
@Brandon Yes, $x$ can be equal to $y$. (The definition doesn't say anything to rule this out.) But Mark is right, $3$ is not idempotent in $\mathbb{Z}/12\mathbb{Z}$.
May
20
comment Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent.
Do your rings have to be commutative? If not, then $2\times2$ matrices over a field will do.
May
20
comment Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent.
If $n$ is prime, then $R$ is a field and so there are no non-zero nilpotents, and no idempotents other than $0$ or $1$. So if there for an example of this form, $n$ is necessarily not prime.
May
20
comment Finding a basis for intersection of two subspaces
A small notational point that lots of people get wrong - the sets you have written are not subspaces. You probably mean that $w_1$ and $w_2$ are the spans of those sets. Also, you should say "a basis", rather than "the basis".
May
13
comment Summation Indices - How to interpret the zero index?
@penland365 That makes sense - if you've actually proved the telescoping sum formula (which essentially must be by induction), then you'd just be repeating that proof in this case if you did induction here, so telescoping does seem to be the intended strategy.
May
12
comment Summation Indices - How to interpret the zero index?
I don't know the book, so can't check if this makes sense - but this looks like an early exercise in using induction. If you haven't covered induction, then the telescoping sum idea is more likely to be the intended approach. In case this is what's confusing you - there is no sense in which $a_0$ is "always" $0$, so there's nothing to prove there. You seem to be making the observation that $\sum_{i=1}^na_i=\sum_{i=0}^na_i$ if you define $a_0$ to be equal to $0$; this is fine. (But maybe you knew this and I misunderstood you.)
May
6
comment Notation in modulo groups
That sounds promising - $\varphi(m)=|\mathbb{Z}_m^*|$.
May
1
comment Prove by induction that $\;A^n = PD^nP^{-1}$
I don't know if you get notified of this automatically - in case not, I've edited my answer to comment on your attempted solution.