Articles of reference request

How to construct magic squares of even order

Could someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?

Generalization of a product measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure on $\mathfrak B(Y)$ and the map $x\mapsto K_x(B)$ is $\mathfrak B(X)$-measurable for any $B\in \mathfrak B(Y)$. Let us further […]

Normal approximation of tail probability in binomial distribution

From the Berry Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|\in O\left(\frac 1{\sqrt n}\right)$$ whereby $B_n$ has the standardized binomial distribution and $N$ has the standardized normal distribution. I can prove this for $x\approx 0$ with Stirling’s formula and a similar proof shown here. Unfortunately Stirling’s approximation becomes worse the bigger $|x|$ is, so […]

If a rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$

Is it true that if a rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$ I would like to know if there is a book where this subject is fully detailed. EDIT: Improve the question!

Connected sums and their homology

Edit: I already received a good answer to my second question. I’d be interested in a hint about the first one, as well. Thanks in advance! I’m interested in compact Riemann surfaces and their homology. In this question, Kundor proposes a nice drawing of the connected sum of tori, saying that it is clearer than […]

Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”

I am doing some tests with strictly increasing integer sequences whose gaps between consecutive elements show a “pseudorandom” behavior, meaning “pseudorandom” that the gaps do not grow up continuously, but they change from a bigger value to a smaller one and vice versa due to the properties of the sequence without an easy way of […]

How did Euler prove the partial fraction expansion of the cotangent function: $\pi\cot(\pi z)=\frac1z+\sum_{k=1}^\infty(\frac1{z-k}+\frac1{z+k})$?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I’ve seen several modern proofs of it and they all seem to rely either on the Herglotz trick or on the residue theorem. I recon Euler had neither nor at his […]

book for metric spaces

Can anybody suggest me a good book on Metric Spaces. Although I am not new to this subject, but want to polish my knowledge. I want a book which can clearly clear my basics. I want to start from the basics. Kindly suggest me. Thanks a lot.

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)

Moving to a conformal metric

Given a generic 2-dimensional metric $$ ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2 $$ what is the change of coordinates that move it into the conformal form $$ ds^2=e^{\phi(\xi,\zeta)}(d\xi^2+d\zeta^2) $$ being $\xi=\xi(x,y)$ and $\zeta=\zeta(x,y)$? Is it generally known? Also a good reference will fit the bill. Thanks beforehand.