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.

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 […]

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 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 […]

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 […]

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers (http://en.wikipedia.org/wiki/Cumulant#Cumulants_of_some_continuous_probability_distributions)?

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: […]

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 […]

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 […]

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} = […]

Intereting Posts

Show that a given set has full measure or measure 0
Polygon Inequality
Formal Power Series — what's in it?
Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.
Matrix equation implies invertibility
Prove that if $\int_{a}^{b} f(x)$ exists, $\delta >0 $ such that $|\sigma_1 -\sigma_2|<\epsilon$
Calculate sum of an infinite series
Compute the minimum distance between the centre to the curve $xy=4$.
Contest problem about convergent series
Uniformly distributed rationals
Are there iterative formulas to find zeta zeros?
Proving that $\int_0^\infty\Big(\sqrt{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$
Looking for an identity for characteristic polynomial of a matrix to the power of n
Formula for the harmonic series $H_n = \sum_{k=1}^n 1/k$ due to Gregorio Fontana
On the number of quadratic residues $\pmod{pq}$ where$p$ and $q$ are odd primes.