Articles of reference request

Encyclopedia of Mathematics? (non-alphabetical)

Do you know any encyclopedia of mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level? And what’s the difference between say, if I am studying calculus from a mathematical encyclopedia and if I am studying calculus from a university based calculus textbook?

Are projective modules over an artinian ring free?

Quoting a comment to this question: By a theorem of Serre, if $R$ is a commutative artinian ring, every projective module [over $R$] is free. (The theorem states that for any commutative noetherian ring $R$ and projective module $P$ [over $R$], if $\operatorname{rank}(P) > \dim(R)$, then there exists a projective [$R$-module] $Q$ with $\operatorname{rank}(Q)=\dim(R)$ such […]

Some questions about Fitting ideals

Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation $$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ideal generated by all the $(n-k)\times (n-k)$ minors of the matrix representation of the map $$ R^{\oplus […]

Regularity up to the boundary

Let $L$ be a second order linear elliptic differential operator on an open bounded subset $U\subset \mathbb R^n$, with smooth uniformly bounded coefficients. Suppose the boundary of $U$ is $C^\infty$. Suppose $f\in C_c^\infty(U)$ ($f$ is smooth and has compact support in $U$). Must there exist a solution $u$ to the PDE $Lu = f$, $u|_{\partial […]

How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam’s original argument about measure theory and measurable cardinals. Here is the result I am looking for: The smallest cardinal $\kappa$ that admits a non-trivial countably-additive two-valued measure must be inaccessible. The original paper can be found below […]

Interchanging the order of limits

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?

Any idea about N-topological spaces?

In Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly introduced the idea of bitopological spaces. Is there any paper concerning the generalization of this concept, i.e. a space with any number of topologies?

Book for Markov Chain Monte Carlo methods

Can anyone recommend a good for MCMC? I have worked with HMMs, Markov Chains in the past but nothing on simulation. So something in the intermediate level would be great. Also, if you know of any introductory books on stochastic simulation, that would be great.

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: $$\frac{1}{x}\left(\sum_{k=1}^xM(k)\right)+2\sim\Im((\text{K}x) ^{\rho_{1}}/\text{c})$$ where $\text{c}\approx64,\ \rm{K}$ is Catalan’s constant, $M(k)$ is the Mertens function of $k$, and $\rho_n$ is the nth zeta zero, and the […]

A (possibly) easier version of Bertrand's Postulate

While attending a math puzzle contest, my friend (a math student) asked me to prove that $$\sum_{k=1}^n \frac{1}{k} \notin \mathbb{Z} \quad \forall n \geq 2$$ Being the first time seeing this problem, I came up with a proof that required the following conjecture: (1) Given a composite number $n \geq 4$, $\exists p$ prime such […]