Articles of soft question

What would happen if we created a vector space over an integral domain/ring.

I’ve always been curious about this, why do we use fields as the only algebraic structure to put vector spaces over? It seems a bit arbitrary to me, so I was wondering what would happen it we replaced the requirement with something less structured like a ring (with unity, we don’t want to violate the […]

Book recommendation for Putnam/Olympiads

I have been concentrating on olympiad questions, and PUTNAM exams, Putnam is my main focus. Can you suggest a book from one of these: Problem Solving Strategies By Arthur Engel Putnam and Beyond by Andreescu Titu et. al First, can you tell me which one is better from the above? I only have one choice? […]

Sphere homeomorphic to plane?

I just took a course in general topology about a month back, and I was wondering whether it was possible to explain why the Earth seems flat from our point of view but is in fact a sphere using the concept of a homeomorphism? Is it the fact that the sphere and plane are homeomorphic […]

Have any one studied this binomial like coefficients before?

Consider the following identities. $\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n}{r}\dbinom{n-r-1}{r-1}$ $\dfrac{n-1}{n-r}\dbinom{n-r}{r-1}+\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n+1}{n+1-r}\dbinom{n+1-r}{r}$ I studied binomial like coefficient and find these two new identities. I want to know has any one studied about the binomial like coefficient $\dfrac{n}{n-r}\dbinom{n-r}{r}$ or is there any combinatorical meaning for this?

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley’s theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, this theorem is taught without applications. I came across one interesting application: If $|G|=2n$ where $n$ is […]

“Mathematical Induction”

I realize this question borders on not qualifying as answerable or mathematical enough, but I would suspect it relevant somehow. I’ll remove it if it’s not. If you look at some explanations of mathematical induction you can find authors first choose to point out that mathematical induction isn’t inductive, in the sense of inductive reasoning, […]

Have Information Theoretic results been used in other branches of mathematics?

consider this a soft-question. Information Theory is fairly young branch of mathematics (60 years). I am interested in question, whether there have been any information theoretic results that had impact on other seemingly ‘unrelated’ branches of mathematics? Looking forward to hearing you thoughts.

I am going to learn these math topics , please suggest me where to start?

I always did poor in mathematics and i even quit my mathematics from 10th grade but since I was good in programming ( C++ and Java) I took course related to computers in my college where I am going to join now. I know Mathematics plays a vital role in programming, I am having below […]

Grothendieck's “Relative” Point of View

I have often read that Grothendieck’s insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why is it important that we study S-schemes, change of base, fibered products, and the like? Are there any specific concrete […]

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set – or even subsets of $\mathbb R$ of a finite measure. They also clearly understtod the proof that there are no ways to put a uniform distribution (as […]