One of the standard definitions of Riemann Integral is as follows: Let $f$ be bounded on $[a, b]$. For any partition $P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ of $[a, b]$ and any choice of points $t_{k} \in [x_{k – 1}, x_{k}]$ the sum $$S(P, f) = \sum_{k = 1}^{n}f(t_{k})(x_{k} – x_{k – 1})$$ is […]

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} \Delta u$ with initial data $u(0,x)=f(x)$, considered on the whole space. Here the solution is given by $u(t,x)=\mathbb{E}(f(x+W_t))$ where $W_t$ is a […]

This question already has an answer here: Intuitive Understanding of the constant “$e$” 18 answers

As I was preparing to spend a month studying p-adic analysis, I realized that I’ve never seen the theory of p-adic numbers applied in other branches of mathematics. I can certainly see that the field $\mathbb Q_p$ has many nice properties, e.g. $X\subset Q_p$ closed and bounded implies $X$ compact, $\sum\limits_{n=0}^\infty x_n$ exists iff $\lim\limits_{n\to\infty}x_n=0$, […]

Does it exist any program (for linux) which can generate a nice Cayley graph of any $\mathbb Z_n$? (If it’s possible to create such a graph at all, that is.) (where perhaps $n ≤ 100$ or something like that)

I found Project Origami: Activities for Exploring Mathematics in my university’s library the other day and quickly FUBAR’d (folded-up beyond all recognition) the couple sheets of paper I had with me at the time. I showed the book to a couple friends who have studied some topology, and we looked at one project with the […]

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them anymore). The author purports to bound the probability of the Goldbach Conjecture being false as $\approx 10^{–150,000,000,000}$ What is mathematics to make of […]

Why do structured sets, like (N, +) often get referred to just by their set? Under this way of speaking, where N denotes the natural numbers, + addition, and * multiplication, (N, +, *) and (N, +) both can get referred to as N. But, due to our ancestors we can readily talk about (N, […]

Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I’ve worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), […]

I came across this site and am wondering if there is a similar page for Mathematics or its sub-areas. Would be very nice if there is one such site which provides ‘canonical’ references for each sub area and preferably is editable like the Wikipedia system so that it reflects entire community’s opinion and not just […]

Intereting Posts

If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
General rule for the integral of $\frac1{x^n}$
$x^4 + y^4 = z^2$
A problem about an $R$-module that is both injective and projective.
Questions about weak derivatives
Why are continuous functions not dense in $L^\infty$?
Finding a Rotation Transformation from two Coordinate Frames in 3-Space
What is step by step logic of pinv (pseudoinverse)?
Existence of an embedding from the rational numbers to $(0,1)$
cosh x inequality
Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} – \det M_{ij}\det M_{ji}$
Isometry <=> Adjoint left inverse
Prove $BA – A^2B^2 = I_n$.
Show $\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}d\theta=0$
Continued fraction estimation of error in Leibniz series for $\pi$.