Articles of reference request

What is the most complete book of integrals and series?

I’m looking for something like “If it’s not in this book, it’s not known”. I’ve got a copy of Gradshteyn and Ryzhik, which seems pretty good. But I’m hoping there are some better ones out there.

Reference request: Introduction to Applied Differential Geometry for Physicists and Engineers

I’m looking for a book on differential geometry or differential topology that is comprehensive and reads at the level of someone with engineering background (i.e. Boyce’s ODE, Stewart’s Calculus, Axler’s Linear algebra). The book should motivate the idea of manifold as it is used in physics and engineering and move up to stuff like vector […]

Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf’s original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, but unfortunately I do not read German. Do you know a translation or a reference following the same argument? Nota Bene: I am aware […]

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then they are of the same length (the sum of the powers of the irreducible factors) and the irreducibles involved are associated to […]

Which isomorphism of coordinate rings corresponds to isomorphisms of affine varieties?

Consider the following claim: Let $X \subset k[x_1,\dots,x_n]$ and $Y\subset k[y_1,\dots,y_m]$ be algebraic sets and suppose that we have a ring isomorphism $\varphi: \mathcal{O}_Y \to \mathcal{O}_X$. Show that the algebraic sets $X$ and $Y$ are isomorphic. Now, I know that if $\varphi$ is an isomorphism of $k$-algebras, then $X$ and $Y$ are isomorphic. And I […]

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers (

Is this equality in a double category true?

Caveat: This is a utterly trivial question from a person who always learned to manipulate diagrams in a double category “from the ground”; I’ll be glad even if you simply address me to any source which gives precise rules of transformations for such diagrams (and this explains the “reference-request” tag). My problem is the following: […]

Essays on the real line?

Are there any essays on real numbers (in general?). Specifically I want to learn more about: The history of (the system of) numbers; their philosophical significance through history; any good essays on their use in physics and the problems of modeling a ‘physical’ line. Cheers. I left this vague as google only supplied Dedekind theory […]

Weakly convex functions are convex

Let us consider a continuous function $f \colon \mathbb{R} \to \mathbb{R}$. Let us call $f$ weakly convex if $$ \int_{-\infty}^{+\infty}f(x)[\varphi(x+h)+\varphi(x-h)-2\varphi(x)]dx\geq 0 \tag{1} $$ for all $h \in \mathbb{R}$ and all $\varphi \in C_0^\infty(\mathbb{R})$ with $\varphi \geq 0$. I was told that $f$ is weakly convex if, and only if, $f$ is convex; although I can […]

Comparing the deviation of a function from its mean on concentric balls

Let $\Omega \subset \mathbb{R}^n$ ($n\geq 2$) a domain with smooth boundary. Consider a ball $B(x_0,R) \Subset \Omega$ . Is true that $$ \int_{B(x_0,r)} | u – u_{x_0 , r}|^p \leq \int_{B(x_0,R)} | u – u_{x_0 , R}|^p,$$ for $u \in W^{1,p}(\Omega)(p >1)$ and for all $0 < r \leq R$, where $u_{x_0 , r} = […]