Divergence theorem on stratified spaces

It is very common in physics and engineering to apply the divergence theorem to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating this very classic theorem shows a boundary of a 3D domain that is stratified (vertices are dimension 0, edges are of dimension … Read more

Verifying a source that lacks a citation

In this German Mathematics Wikibook page, formula 0.5 lists the following equation ∫10sin(πx)xx(1−x)1−x dx=πe24 as supposedly attributed to Ramanujan (Google translate gives “formula by Ramanujan”). However, the page lacks a citation section, which is absolutely unfortunate, so I’m not certain whether to trust this source or not. How would I be able to verify the source … Read more

Generalize $\frac1{48^{1/4}\,K(k_3)}\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+27x^3}}=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27\big)=\frac47\,$?

These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of, $${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}\tag1$$ The function $\eta(\tau)$ below is the Dedekind eta function. I. Case $a=\tfrac13$ Conjecture: There are an infinitely many algebraic numbers $\alpha, \beta$ such that $$H_1(\tau) =\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-\alpha \big)=\beta$$ given by, $$\alpha = \frac1{4\sqrt{27}}\big(\lambda^3-\sqrt{27}\,\lambda^{-3}\big)^2$$ where $\lambda=\large{\frac{\eta(\frac{\tau+1}3)}{\eta(\tau)}}$ and $\tau=\frac{1+N\sqrt{-3}}2$ for any integer $N>1$. Examples: $$H_1\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-4 \big)=\tfrac3{5^{5/6}}$$ $$H_1\big(\tfrac{1+7\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27 … Read more

English language and Mathematics

I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question. Let $\mathcal M$ be a smooth manifold, let $X$ be a smooth vector field on $\mathcal M$ and let $\Sigma$ be a smooth hypersurface of $\mathcal M$. Let … Read more

Fractional integral inequality (Hardy-Littlewood-Sobolev)

I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y – x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q < \infty $, and $\mu = 1 + \frac{1}{q} – \frac{1}{p}$. For the integral \begin{equation} I(x) = \int_{\mathbb{R}} \frac{f(y) }{|y – x|^{\mu}} \, dy \end{equation} there … Read more

Is the following integral positive or not?

Let n be a given even positive integer. We have the following integral ∫10⋯∫10n∏i=1n∏j=1(xi−yj)dx1⋯dxndy1⋯dyn=∫10⋯∫10(∫10n∏j=1(x−yj)dx)ndy1⋯dyn>0. Let’s consider a similar integral where n is also an even positive integer: An=∫10⋯∫10∫2π0⋯∫2π0n∏i=1n∏j=1(xi−yj+icos(αi−βj))dx1⋯dxndy1⋯dyndα1⋯dαndβ1⋯dβn. It is easy to see that An is a real number for every n. My question is whether An is positive or not. Answer AttributionSource : Link … Read more

$L^2$-valued integral as parameter integral

Setting Let us regard the Hilbert space $L^2(0,1)$ and the $C_0$-semigroup $(T(t))_{t\geq 0}$ defined by $$ T(t):\left\{ \begin{array}{rml} L^2(0,1) & \to & L^2(0,1), \\ [f]_{\sim} &\mapsto &\left[x \mapsto \begin{cases} f(x+t), & \text{if}\; x+t<1\\ 0, & \text{else} \end{cases} \right]_{\sim}. \end{array} \right. $$ It is easy to verify that this is indeed a $C_0$-semigroup. Therefore, the mapping … Read more

Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that computes the volume of $C\cap B(0,1)$ to arbitrary precision? $C$ has $d$ different classes of edges and many edges of each class, so it can be very … Read more

Integration on a family of differential forms

Let X be a smooth manifold, and denote by Ω∗(X) the set of all smooth differential forms on X. Assume we have a family of differential forms ωt∈Ω∗(X), t∈E, parameterized by a compact set E⊂RN. Let’s say the map t↦ωt is smooth. Then is there a natural way to talk about the ‘average’ or ‘integration’ … Read more

Can this integral be made nonpositive?

Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \mathbb{S}^3$ such that $-p_0 \not \in M$, and define a function $c_{p_0} : M \to \mathbb{R}$ by $$c_{p_0}(p) = \frac{\langle \eta(p), p_0 \rangle}{1 … Read more