23,607 reputation
32870
bio website maths.usyd.edu.au/ut/…
location Sydney, Australia
age 22
visits member for 2 years, 11 months
seen 1 hour ago

I am in my fourth year studying mathematics at the University of Sydney. In addition to learning math, I enjoy helping others with it (hence my membership here) and watching/playing basketball.


2d
comment How do we know that $\mathbb{Q}$ is the initial field of characteristic $0$?
Every field of characteristic 0 has a copy of Q embedded in it. Any field map Q to F must send 1 to 1, and then the rest is determined.
2d
comment Contraction mapping principle application
@Nick Differential.
2d
comment Contraction mapping principle application
@Nick The Mean Value Theorem can help you here.
Jul
22
comment How much math is there?
A suggested measure of how much math is "known": How many of the worlds most knowledgeable mathematicians would we have to collect so that 9 of any 10 known pieces of mathematics (a publication, private communications, folklore results, everything) would be familiar to at least one of them, or that they could prove within two hours from what they already know. Some guesstimates: In 1900, perhaps a dozen. Today, at least 100.
Jul
22
comment Exercises in category theory for a non-working mathematican (undergrad)
+1. Just from what I can see off the Amazon preview, the book looks excellently written at an elementary level. A comment for beginners: Do not be put off that Leinster's book "only" covers (Co)Limits, Representables and Adjoints. Even though Mac Lane covers much more, a solid foundation in just these three concepts will take you surprisingly far. It's similar to Set Theory: In an elementary text of 10 chapters, you'll use things from the first 5 chapters every day of your life, and things from the last 5 chapters only on occasion (depending on what you study, of course).
Jul
22
comment Is it necessary to know all the details in proofs of theorems you study in PDE's?
See this math.SE thread.
Jul
22
revised Why do only fixed points contribute to the Euler characteristic?
added 5728 characters in body
Jul
22
comment Why do only fixed points contribute to the Euler characteristic?
@Brenin Sorry, I am not aware of a result in the generality you want, and if the general result is indeed true, then its proof is probably more difficult than the one attempted in your OP. In every instance I've seen a result of this type, details specific to the situation need to be involved to push through an argument of the form user160609 suggested. I'll update my answer to include some more general facts to help you do this in situations you encounter.
Jul
22
comment Continuity of a function in two variables
@Genomeme Yes, that's correct.
Jul
21
comment If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?
Non-zero prime ideals in PIDs are maximal =]
Jul
21
answered On a basic tensor product question
Jul
19
answered prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.
Jul
19
answered Why do only fixed points contribute to the Euler characteristic?
Jul
18
comment Proving this corollary of the Unique Factorization Theorem (of Integers)…
Hint: You have a good suspicion of what $d$ should be ($k_i=\text{min}(n_i,m_i)$). Now just verify that this number satisfies the properties of the gcd. So show that it divides $a$ and $b$, and then show that any common divisor of $a$ and $b$ divides $d.$
Jul
17
awarded  Revival
Jul
16
answered Continuity of a function in two variables
Jul
14
comment My notes say a torus and a sphere homeomorphic?
Your definition of the torus is the standard one, and indeed a torus is not homeomorphic to a sphere. The easiest way to see this is via the Fundamental Group: Any loop on a sphere contracts to a point while you can find a loop on the torus that does not contract to a point. Some other ways are via Euler Characteristic or Homology groups. Unfortunately a justification by point-set topology isn't coming to mind at the moment.
Jul
14
awarded  Informed
Jul
14
answered there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$
Jul
14
comment there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$
You should be precise about what type of objects/homomorphisms these are. Abelian groups under addition, rings, what?