I dread mathematics, and I believe it’s because I have come to associate mathematics with the experience of terrible teachers. All of my math teachers have been grumpy, but one in particular was the epitome of evil. She would take any opportunity to yell and scream at me when I struggled to comprehend the problems […]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The idea is like this: consider $T = \{ (x,y,z) \mid x + y + z = 1, […]

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used “Introduction to Analysis” by Gaughan. While it’s a good book, I’m not sure it’s suited for self study by itself. I […]

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference “Why this is hard ?” Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but ultimately failed to conclude this.

I’m starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia’s notes, but I found them a bit too “logical” so to say. Are they good? What else do you recommend?

Okay so I understand what calculus, linear algebra, combinatorics and even topology try to answer, but why invent category theory? In wikipedia it says it is to formalize. As far as I can tell it sort of like generalizes a bunch of fields in mathematics, like in topology, graphs and groups we have isomorphisms and […]

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global sections ect) but I am missing the bigger picture. Why should I care if a functor […]

How to to calculate the maximim or minimum of two numbers without using “if” ( or something equivalant to that manner)? The above question is often asked in introductory computer science courses and is answered using this method. Now although it is not obvious, but using absolute value is also equivalant to using an if […]

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group (since generally, any set can be given the discrete topology), but there are groups that have no smooth structure. […]

When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} (2 + 3i) = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + \sum\limits_{i=1}^{n}3i$$ However, I don’t think I know all the useful shortcuts here. Are there other computational tricks one should be aware of? What’s a good way for thinking about […]

Intereting Posts

Is there a “deep line” topological space in analogue to the “long line” $\omega_1\times[0,1)$?
Integrating $\frac{\log(1+x)}{1+x^2}$
Application of Luca's theorem
A nice introduction to forcing
What am I doing wrong in these quartic formula calculations?
Can one tell based on the moments of the random variable if it is continuos or not
Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?
Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
If $a_1,\ldots,a_n>0$ and $a_1+\cdots+a_n<\frac{1}{2}$, then $(1+a_1)\cdots(1+a_n)<2$.
$x^5 – y^2 = 4$ has no solution mod $m$
Lindelöf'ize a space?
A question on Terence Tao's representation of Peano Axioms
What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$?
Does absolute convergence of a sum imply uniform convergence?
bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$