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?

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 […]

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 […]

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 […]

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 […]

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

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?

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.

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 […]

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 […]

Intereting Posts

Partial derivative of integral: Leibniz rule?
Differential of transposed matrices
What Topology does a Straight Line in the Plane Inherit as a Subspace of $\mathbb{R_l} \times \mathbb{R}$ and of $\mathbb{R_l} \times \mathbb{R_l}$
Formula for the sum of $\ n\cdot 1 + (n-1)\cdot 2 + … + 2 \cdot (n-1) + 1\cdot n$
Calculate distance in 3D space
Distance from $x^n$ to lesser polynomials
Strictly associative coproducts
Prove that $\displaystyle \sum_{1\leq k<j\leq n} \tan^2\left(\frac{k\pi}{2n+1}\right)\tan^2\left(\frac{j\pi}{2n+1}\right)=\binom{2n+1}{4} $
Cauchy functional equation with non choice
Is there a bijection between the reals and naturals?
Convergence in measure implies convergence almost everywhere (on a countable set!)
Circular determinant problem
Three linked question on non-negative definite matrices.
Why does an argument similiar to 0.999…=1 show 999…=-1?
Is the polynomial $x^{105} – 9$ reducible over $\mathbb{Z}$?