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Good books/resources for learning how to visualize Mobius transformations

Posts like this one have always baffled me. I have taken a few complex variables courses in my time and it seems that every time I take a course like this, either the textbook isn't great at explaining...

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Generalising a dot-in-circle puzzle's solution.

Motivation:Puzzle: Draw $\odot$ on paper without lifting your pen.Solution: Start by putting your pen in the centre of the paper to create the dot. Then fold the paper such that the edge of it touches...

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An approximation to $\ln(1+e^x)$ and how to use it for splitting the...

Trying to split the logarithm of the sum of two exponential functions (this question), I found the following approximation for the Softplus function$f(x)=\ln(1+e^x)$:$$\ln(1+e^x) \approx \begin{cases}...

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What is the most influential work of Grothendieck in mathematics?

Recently Alexander Grothendieck has passed away but his mathematical wave is still alive and passes its growth ages. It is hard to describe the influence of such a great man in mathematics just in few...

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What knots exist on $M$?

Consider mutually transversal copies $\zeta_k=(\Bbb H^2 \times \Bbb R)_k$ each geodesically compactified and consider $M=\bigcup_{k=1}^4 \partial \zeta_k$. What knots are permissible on $M$? I note...

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Why points are defined without =?

In so many math books we find definitions like this:Let $A(2,-3)$ be a point...Why the sign '$=$' is omitted for points? Where this habit comes from? Shouldn't we write 'Let $A=(2,-3)$ be a point' ?I...

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Can the Bourbaki series be used profitably by undergraduates?

Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the...

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Are there any mathematical modelling that doesn't use differential equations?...

What kind of math can model phenomena without using ODE/PDE ?

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Companions to Rudin?

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...

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Partial Derivatives in polar coordinates

Thomas's Calculus covers double integration in polar coordinates(Section 15.4), but does not cover partial derivatives in polar coordinates, except for a single question in the exercises which asks for...

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Chess and mathematics

I have to choose a research-like project to follow the next year. Because I'm a chess enthusiast, I was thinking of trying to tackle an (open) problem related to chess, and relevant to mathematics. Can...

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Is there a setup in which is the Woodbury formula actually useful?

Let $A, C, U, V$ denote matrices of suitable sizes and let $A, C$ be invertible. Then Woodbury formula states that$$ \left(A + UCV \right)^{-1} = A^{-1} - A^{-1}U \left(C^{-1} + VA^{-1}U \right)^{-1}...

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Math Diagram Drawing

I recently saw this old sketch of the Gamma function that was done by hand and tools, and I thought that it was really cool. I realized before computer simulations and graphics, if we wanted to...

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Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we...

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Can the natural logarithm be meaningfully "natural" in the category theory...

This may be more of a pun than anything else, as the two usages of the word "natural" are unrelated, but I was wondering if there was any nontrivial sense in which the natural logarithm could be viewed...

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Are these test paper questions as unclear and ambiguous as I think they are?

The Local Education Authority here in Wales are running a free voluntary course online for maths dunces like me to help us to help our 7-14 year olds with maths.There's a free optional paper leading to...

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Impact of homotopical algebra on homological agebra

I was recently reading through Dwyer and Spalinski's notes introducing model categories. In it is shown a different way to describe $Ext(A,B)$ in homotopical terms (see proposition 7.3, I am not...

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Artificial Smoothing, Diffusion and other effects of numerical Schemes

I am looking for an explanation for the terminology "artificial smoothing" and "artificial diffusion" in the context of numerical schemes.For instance, consider the upwind scheme where we estimate a...

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Example of Complex Pythagorean Triples

I am looking for example of a Pythagorean Triple with Gaussian Integers.I followed the links and looked at followings :Relation to Gaussian integers...

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Why is the Daniell integral not so popular?

The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some...

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