Articles of soft question

Learning differential calculus through infinitesimals

In class, we’ve studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and then integrals. But the teacher really did everything to avoid talking about infinitessimals. For instance when we talked about variable changes we had to swallow that for a […]

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I am currently a masteral student of mathematical finance, expecting to graduate sometime this year. I am not particularly interested in mathematical finance anymore […]

Mnemonics for linear algebra

Sometimes formulas in linear algebra are not easy to remember. Some usefulness for the process of remembering can provide application of mnemonics. Do you know some useful mnemonics for this purpose? I’ll give two examples: For the process of finding the inverse of matrix it could be used mnemonic Detminstra what can be translated as […]

What do we mean by an “Elegant Proof”?

What do we mean when we say that a mathematical proof is elegant? Of course one can say that the proof is beautiful, but what do we precisely mean when we say that a proof is beautiful ? Is there a precise way to measure the elegance of a mathematical proof ? I have thought […]

Hairy Ball theorem and its applications

While searching a question about fibre bundles, which was asked here, i got directed to Vector bundles. I noticed this word “Hairy Ball” which sounded eccentric and made a search at Wikipedia. How is the hairy ball theorem related to this statement: You can’t comb the hair on a coconut.

parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which doesn’t lay on a given line, and in spherical geometry, we have no parallels at all. In both of these geometries, we have some kind […]

What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that: An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),……)=1$ generates for natural arguments infinitely many prime numbers. Wikipedia article claims that this is an important open problem. My question is: how important do you consider the answer to this problem, and why?

Is it possible to practice mental math too often?

I’m far from a mathematician, but the field I’m trying to break into (management consulting) requires a fair amount of mental arithmetic. I’m okay, but I’m not even close to as good as I need to be in terms of both speed and accuracy. I have math apps on my iPhone. I use online mental […]

Would nonmath students be able to understand this?

For a course, I am required to do a presentation. The topic could either be something mundane, like a career strategy report, or something more interesting, such as a controversial topic, or an exposition on something you find interesting. What I would like to do is to present math in a way that probably no […]

Why Study Critical Polynomials?

In dynamical systems, I often read about the post-critical orbits. As in take a moduli space of functions $f$ which are self maps. Find general critical points, and see where they orbit. They would then be polynomials in some variables if we allow $f$ to be parameterised. Those are called critical polynomials. It could go […]